Permutation groups

A permutation group is a finite group G whose elements are permutations of a given finite set X (i.e., bijections X \longrightarrow X) and whose group operation is the composition of permutations. The number of elements of X is called the degree of G.

In Sage, a permutation is represented as either a string that defines a permutation using disjoint cycle notation, or a list of tuples, which represent disjoint cycles. That is:

(a,...,b)(c,...,d)...(e,...,f)  <--> [(a,...,b), (c,...,d),..., (e,...,f)]
                  () = identity <--> []

You can make the “named” permutation groups (see permgp_named.py) and use the following constructions:

  • permutation group generated by elements,
  • direct_product_permgroups, which takes a list of permutation groups and returns their direct product.

JOKE: Q: What’s hot, chunky, and acts on a polygon? A: Dihedral soup. Renteln, P. and Dundes, A. “Foolproof: A Sampling of Mathematical Folk Humor.” Notices Amer. Math. Soc. 52, 24-34, 2005.

AUTHORS:

  • David Joyner (2005-10-14): first version
  • David Joyner (2005-11-17)
  • William Stein (2005-11-26): rewrite to better wrap Gap
  • David Joyner (2005-12-21)
  • William Stein and David Joyner (2006-01-04): added conjugacy_class_representatives
  • David Joyner (2006-03): reorganization into subdirectory perm_gps; added __contains__, has_element; fixed _cmp_; added subgroup class+methods, PGL,PSL,PSp, PSU classes,
  • David Joyner (2006-06): added PGU, functionality to SymmetricGroup, AlternatingGroup, direct_product_permgroups
  • David Joyner (2006-08): added degree, ramification_module_decomposition_modular_curve and ramification_module_decomposition_hurwitz_curve methods to PSL(2,q), MathieuGroup, is_isomorphic
  • Bobby Moretti (2006)-10): Added KleinFourGroup, fixed bug in DihedralGroup
  • David Joyner (2006-10): added is_subgroup (fixing a bug found by Kiran Kedlaya), is_solvable, normalizer, is_normal_subgroup, Suzuki
  • David Kohel (2007-02): fixed __contains__ to not enumerate group elements, following the convention for __call__
  • David Harvey, Mike Hansen, Nick Alexander, William Stein (2007-02,03,04,05): Various patches
  • Nathan Dunfield (2007-05): added orbits
  • David Joyner (2007-06): added subgroup method (suggested by David Kohel), composition_series, lower_central_series, upper_central_series, cayley_table, quotient_group, sylow_subgroup, is_cyclic, homology, homology_part, cohomology, cohomology_part, poincare_series, molien_series, is_simple, is_monomial, is_supersolvable, is_nilpotent, is_perfect, is_polycyclic, is_elementary_abelian, is_pgroup, gens_small, isomorphism_type_info_simple_group. moved all the”named” groups to a new file.
  • Nick Alexander (2007-07): move is_isomorphic to isomorphism_to, add from_gap_list
  • William Stein (2007-07): put is_isomorphic back (and make it better)
  • David Joyner (2007-08): fixed bugs in composition_series, upper/lower_central_series, derived_series,
  • David Joyner (2008-06): modified is_normal (reported by W. J. Palenstijn), and added normalizes
  • David Joyner (2008-08): Added example to docstring of cohomology.
  • Simon King (2009-04): __cmp__ methods for PermutationGroup_generic and PermutationGroup_subgroup
  • Nicolas Borie (2009): Added orbit, transversals, stabiliser and strong_generating_system methods

REFERENCES:

  • Cameron, P., Permutation Groups. New York: Cambridge University Press, 1999.
  • Wielandt, H., Finite Permutation Groups. New York: Academic Press, 1964.
  • Dixon, J. and Mortimer, B., Permutation Groups, Springer-Verlag, Berlin/New York, 1996.

Note

Though Suzuki groups are okay, Ree groups should not be wrapped as permutation groups - the construction is too slow - unless (for small values or the parameter) they are made using explicit generators.

sage.groups.perm_gps.permgroup.PermutationGroup(gens=None, gap_group=None, canonicalize=True)

Return the permutation group associated to x (typically a list of generators).

INPUT:

  • gens - list of generators (default: None)
  • gap_group - a gap permutation group (default: None)
  • canonicalize - bool (default: True); if True, sort generators and remove duplicates

OUTPUT:

  • A permutation group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]

We can also make permutation groups from PARI groups:

sage: H = pari('x^4 - 2*x^3 - 2*x + 1').polgalois()
sage: G = PariGroup(H, 4); G            
PARI group [8, -1, 3, "D(4)"] of degree 4
sage: H = PermutationGroup(G); H          # optional - database_gap
Transitive group number 3 of degree 4
sage: H.gens()                            # optional - database_gap
[(1,2,3,4), (1,3)]

We can also create permutation groups whose generators are Gap permutation objects:

sage: p = gap('(1,2)(3,7)(4,6)(5,8)'); p
(1,2)(3,7)(4,6)(5,8)
sage: PermutationGroup([p])
Permutation Group with generators [(1,2)(3,7)(4,6)(5,8)]

There is an underlying gap object that implements each permutation group:

sage: G = PermutationGroup([[(1,2,3,4)]])
sage: G._gap_()
Group( [ (1,2,3,4) ] )
sage: gap(G)
Group( [ (1,2,3,4) ] )
sage: gap(G) is G._gap_()
True
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: current_randstate().set_seed_gap()
sage: G._gap_().DerivedSeries()
[ Group( [ (3,4), (1,2,3)(4,5) ] ), Group( [ (1,5)(3,4), (1,5)(2,4), (1,5,3) ] ) ]

TESTS:

sage: PermutationGroup(SymmetricGroup(5))
Traceback (most recent call last):
...
TypeError: gens must be a tuple, list, or GapElement
class sage.groups.perm_gps.permgroup.PermutationGroup_generic(gens=None, gap_group=None, canonicalize=True)

Bases: sage.groups.group.Group

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G
Permutation Group with generators [(3,4), (1,2,3)(4,5)]
sage: G.center()
Permutation Group with generators [()]
sage: G.group_id()          # optional - database_gap
[120, 34]
sage: n = G.order(); n
120
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: TestSuite(G).run()
center()

Return the subgroup of elements that commute with every element of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3,4)]])
sage: G.center()
Permutation Group with generators [(1,2,3,4)]
sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
sage: G.center()
Permutation Group with generators [()]
centralizer(g)

Returns the centralizer of g in self.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: g = G([(1,3)])
sage: G.centralizer(g)
Permutation Group with generators [(2,4), (1,3)]
sage: g = G([(1,2,3,4)])
sage: G.centralizer(g)
Permutation Group with generators [(1,2,3,4)]
sage: H = G.subgroup([G([(1,2,3,4)])])
sage: G.centralizer(H)
Permutation Group with generators [(1,2,3,4)]
character(values)

Returns a group character from values, where values is a list of the values of the character evaluated on the conjugacy classes.

EXAMPLES:

sage: G = AlternatingGroup(4)
sage: n = len(G.conjugacy_classes_representatives())
sage: G.character([1]*n)
Character of Alternating group of order 4!/2 as a permutation group
character_table()

Returns the matrix of values of the irreducible characters of a permutation group G at the conjugacy classes of G. The columns represent the conjugacy classes of G and the rows represent the different irreducible characters in the ordering given by GAP.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: G.order()
12
sage: G.character_table()
[         1          1          1          1]
[         1          1 -zeta3 - 1      zeta3]
[         1          1      zeta3 -zeta3 - 1]
[         3         -1          0          0]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3)]])
sage: CT = gap(G).CharacterTable()

Type print gap.eval("Display(%s)"%CT.name()) to display this nicely.

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: G.order()
8
sage: G.character_table()
[ 1  1  1  1  1]
[ 1 -1 -1  1  1]
[ 1 -1  1 -1  1]
[ 1  1 -1 -1  1]
[ 2  0  0  0 -2]
sage: CT = gap(G).CharacterTable()

Again, type print gap.eval("Display(%s)"%CT.name()) to display this nicely.

sage: SymmetricGroup(2).character_table()
[ 1 -1]
[ 1  1]
sage: SymmetricGroup(3).character_table()
[ 1 -1  1]
[ 2  0 -1]
[ 1  1  1]
sage: SymmetricGroup(5).character_table()
[ 1 -1  1  1 -1 -1  1]
[ 4 -2  0  1  1  0 -1]
[ 5 -1  1 -1 -1  1  0]
[ 6  0 -2  0  0  0  1]
[ 5  1  1 -1  1 -1  0]
[ 4  2  0  1 -1  0 -1]
[ 1  1  1  1  1  1  1]
sage: list(AlternatingGroup(6).character_table())
[(1, 1, 1, 1, 1, 1, 1), (5, 1, 2, -1, -1, 0, 0), (5, 1, -1, 2, -1, 0, 0), (8, 0, -1, -1, 0, zeta5^3 + zeta5^2 + 1, -zeta5^3 - zeta5^2), (8, 0, -1, -1, 0, -zeta5^3 - zeta5^2, zeta5^3 + zeta5^2 + 1), (9, 1, 0, 0, 1, -1, -1), (10, -2, 1, 1, 0, 0, 0)]

Suppose that you have a class function f(g) on G and you know the values v_1, \ldots, v_n on the conjugacy class elements in conjugacy_classes_representatives(G) = [g_1, \ldots, g_n]. Since the irreducible characters \rho_1, \ldots, \rho_n of G form an E-basis of the space of all class functions (E a “sufficiently large” cyclotomic field), such a class function is a linear combination of these basis elements, f = c_1 \rho_1 + \cdots + c_n \rho_n. To find the coefficients c_i, you simply solve the linear system character_table_values(G) [v_1, ..., v_n] = [c_1, ..., c_n], where [v_1, \ldots, v_n] = character_table_values(G) ^{(-1)}[c_1, ..., c_n].

AUTHORS:

  • David Joyner and William Stein (2006-01-04)
cohomology(n, p=0)

Computes the group cohomology H^n(G, F), where F = \ZZ if p=0 and F = \ZZ / p \ZZ if p > 0 is a prime. Wraps HAP’s GroupHomology function, written by Graham Ellis.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(4)
sage: G.cohomology(1,2)                            # optional - gap_packages
Multiplicative Abelian Group isomorphic to C2
sage: G = SymmetricGroup(3)
sage: G.cohomology(5)                              # optional - gap_packages
Trivial Abelian Group
sage: G.cohomology(5,2)                            # optional - gap_packages
Multiplicative Abelian Group isomorphic to C2
sage: G.homology(5,3)                              # optional - gap_packages
Trivial Abelian Group
sage: G.homology(5,4)                              # optional - gap_packages
Traceback (most recent call last):
...
ValueError: p must be 0 or prime

This computes H^4(S_3, \ZZ) and H^4(S_3, \ZZ / 2 \ZZ), respectively.

AUTHORS:

  • David Joyner and Graham Ellis

REFERENCES:

cohomology_part(n, p=0)

Computes the p-part of the group cohomology H^n(G, F), where F = \ZZ if p=0 and F = \ZZ / p \ZZ if p > 0 is a prime. Wraps HAP’s Homology function, written by Graham Ellis, applied to the p-Sylow subgroup of G.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.cohomology_part(7,2)                   # optional - gap_packages
Multiplicative Abelian Group isomorphic to C2 x C2 x C2
sage: G = SymmetricGroup(3)
sage: G.cohomology_part(2,3)                   # optional - gap_packages
Multiplicative Abelian Group isomorphic to C3

AUTHORS:

  • David Joyner and Graham Ellis
commutator(other=None)

Returns the commutator subgroup of a group, or of a pair of groups.

INPUT:

  • other - default: None - a permutation group.

OUTPUT:

Let G denote self. If other is None then this method returns the subgroup of G generated by the set of commutators,

\{[g_1,g_2]\vert g_1, g_2\in G\} = \{g_1^{-1}g_2^{-1}g_1g_2\vert g_1, g_2\in G\}

Let H denote other, in the case that it is not None. Then this method returns the group generated by the set of commutators,

\{[g,h]\vert g\in G\, h\in H\} = \{g^{-1}h^{-1}gh\vert  g\in G\, h\in H\}

The two groups need only be permutation groups, there is no notion of requiring them to explicitly be subgroups of some other group.

Note

For the identical statement, the generators of the returned group can vary from one execution to the next.

EXAMPLES:

sage: G = DiCyclicGroup(4)
sage: G.commutator()
Permutation Group with generators [(1,3,5,7)(2,4,6,8)(9,11,13,15)(10,12,14,16)]

sage: G = SymmetricGroup(5)
sage: H = CyclicPermutationGroup(5)
sage: C = G.commutator(H)
sage: C.is_isomorphic(AlternatingGroup(5))
True

An abelian group will have a trivial commutator.

sage: G = CyclicPermutationGroup(10)
sage: G.commutator()
Permutation Group with generators [()]

The quotient of a group by its commutator is always abelian.

sage: G = DihedralGroup(20)
sage: C = G.commutator()
sage: Q = G.quotient(C)
sage: Q.is_abelian()
True

When forming commutators from two groups, the order of the groups does not matter.

sage: D = DihedralGroup(3)
sage: S = SymmetricGroup(2)
sage: C1 = D.commutator(S); C1
Permutation Group with generators [(1,2,3)]
sage: C2 = S.commutator(D); C2
Permutation Group with generators [(1,3,2)]
sage: C1 == C2
True

This method calls two different functions in GAP, so this tests that their results are consistent. The commutator groups may have different generators, but the groups are equal.

sage: G = DiCyclicGroup(3)
sage: C = G.commutator(); C
Permutation Group with generators [(5,7,6)]
sage: CC = G.commutator(G); CC
Permutation Group with generators [(5,6,7)]
sage: C == CC
True

The second group is checked.

sage: G = SymmetricGroup(2)
sage: G.commutator('junk')
Traceback (most recent call last):
...
TypeError: junk is not a permutation group
composition_series()

Return the composition series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

sage: set_random_seed(0)
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.composition_series()  # random output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,3), (1,5,4)], Permutation Group with generators [()]]
sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: CS = G.composition_series()
sage: CS[3]
Permutation Group with generators [()]
conjugacy_classes_representatives()

Returns a complete list of representatives of conjugacy classes in a permutation group G. The ordering is that given by GAP.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: cl = G.conjugacy_classes_representatives(); cl
[(), (2,4), (1,2)(3,4), (1,2,3,4), (1,3)(2,4)]
sage: cl[3] in G
True
sage: G = SymmetricGroup(5)
sage: G.conjugacy_classes_representatives ()
[(), (1,2), (1,2)(3,4), (1,2,3), (1,2,3)(4,5), (1,2,3,4), (1,2,3,4,5)]

AUTHORS:

  • David Joyner and William Stein (2006-01-04)
conjugacy_classes_subgroups()

Returns a complete list of representatives of conjugacy classes of subgroups in a permutation group G. The ordering is that given by GAP.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: cl = G.conjugacy_classes_subgroups()
sage: cl
[Permutation Group with generators [()],
 Permutation Group with generators [(1,2)(3,4)],
 Permutation Group with generators [(1,3)(2,4)],
 Permutation Group with generators [(2,4)],
 Permutation Group with generators [(1,4)(2,3), (1,2)(3,4)],
 Permutation Group with generators [(1,3)(2,4), (2,4)],
 Permutation Group with generators [(1,3)(2,4), (1,2,3,4)],
 Permutation Group with generators [(1,3)(2,4), (1,2)(3,4), (1,2,3,4)]]
sage: G = SymmetricGroup(3)
sage: G.conjugacy_classes_subgroups()
[Permutation Group with generators [()],
 Permutation Group with generators [(2,3)],
 Permutation Group with generators [(1,2,3)],
 Permutation Group with generators [(1,3,2), (1,2)]]

AUTHORS:

  • David Joyner (2006-10)
conjugate(g)

Returns the group formed by conjugating self with g.

INPUT:

  • g - a permutation group element, or an object that converts to a permutation group element, such as a list of integers or a string of cycles.

OUTPUT:

If self is the group denoted by H, then this method computes the group

g^{-1}Hg = \{g^{-1}hg\vert h\in H\}

which is the group H conjugated by g.

There are no restrictions on self and g belonging to a common permutation group, and correspondingly, there is no relationship (such as a common parent) between self and the output group.

EXAMPLES:

sage: G = DihedralGroup(6)
sage: a = PermutationGroupElement("(1,2,3,4)")
sage: G.conjugate(a)
Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)]

The element performing the conjugation can be specified in several ways.

sage: G = DihedralGroup(6)
sage: strng = "(1,2,3,4)"
sage: G.conjugate(strng)
Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)]
sage: G = DihedralGroup(6)
sage: lst = [2,3,4,1]
sage: G.conjugate(lst)
Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)]
sage: G = DihedralGroup(6)
sage: cycles = [(1,2,3,4)]
sage: G.conjugate(cycles)
Permutation Group with generators [(1,5,6,2,3,4), (1,4)(2,6)(3,5)]

Conjugation is a group automorphism, so conjugate groups will be isomorphic.

sage: G = DiCyclicGroup(6)
sage: G.degree()
11
sage: cycle = [i+1 for i in range(1,11)] + [1]
sage: C = G.conjugate(cycle)
sage: G.is_isomorphic(C)
True

The conjugating element may be from a symmetric group with larger degree than the group being conjugated.

sage: G = AlternatingGroup(5)
sage: G.degree()
5
sage: g = "(1,3)(5,6,7)"
sage: H = G.conjugate(g); H
Permutation Group with generators [(1,4,6,3,2), (1,4,6)]
sage: H.degree()
6

The conjugating element is checked.

sage: G = SymmetricGroup(3)
sage: G.conjugate("junk")
Traceback (most recent call last):
...
TypeError: junk does not convert to a permutation group element
construction()

EXAMPLES:

sage: P1 = PermutationGroup([[(1,2)]])
sage: P1.construction()
(PermutationGroupFunctor[(1,2)], Permutation Group with generators [()])

sage: PermutationGroup([]).construction() is None
True

This allows us to perform computations like the following:

sage: P1 = PermutationGroup([[(1,2)]]); p1 = P1.gen()
sage: P2 = PermutationGroup([[(1,3)]]); p2 = P2.gen()
sage: p = p1*p2; p
(1,2,3)
sage: p.parent()
Permutation Group with generators [(1,2), (1,3)]
cosets(S, side='right')

Returns a list of the cosets of S in self.

INPUT:

  • S - a subgroup of self. An error is raised if S is not a subgroup.
  • side - default: ‘right’ - determines if right cosets or left cosets are returned. side refers to where the representative is placed in the products forming the cosets and thus allowable values are only ‘right’ and ‘left’.

OUTPUT:

A list of lists. Each inner list is a coset of the subgroup in the group. The first element of each coset is the smallest element (based on the ordering of the elements of self) of all the group elements that have not yet appeared in a previous coset. The elements of each coset are in the same order as the subgroup elements used to build the coset’s elements.

As a consequence, the subgroup itself is the first coset, and its first element is the identity element. For each coset, the first element listed is the element used as a representative to build the coset. These representatives form an increasing sequence across the list of cosets, and within a coset the representative is the smallest element of its coset (both orderings are based on of the ordering of elements of self).

In the case of a normal subgroup, left and right cosets should appear in the same order as part of the outer list. However, the list of the elements of a particular coset may be in a different order for the right coset versus the order in the left coset. So, if you check to see if a subgroup is normal, it is necessary to sort each individual coset first (but not the list of cosets, due to the ordering of the representatives). See below for examples of this.

Note

This is a naive implementation intended for instructional purposes, and hence is slow for larger groups. Sage and GAP provide more sophisticated functions for working quickly with cosets of larger groups.

EXAMPLES:

The default is to build right cosets. This example works with the symmetry group of an 8-gon and a normal subgroup. Notice that a straight check on the equality of the output is not sufficient to check normality, while sorting the individual cosets is sufficient to then simply test equality of the list of lists. Study the second coset in each list to understand the need for sorting the elements of the cosets.

sage: G = DihedralGroup(8)
sage: quarter_turn = G.list()[5]; quarter_turn
(1,3,5,7)(2,4,6,8)
sage: S = G.subgroup([quarter_turn])
sage: rc = G.cosets(S); rc
[[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)],
 [(2,8)(3,7)(4,6), (1,7)(2,6)(3,5), (1,5)(2,4)(6,8), (1,3)(4,8)(5,7)],
 [(1,2)(3,8)(4,7)(5,6), (1,8)(2,7)(3,6)(4,5), (1,6)(2,5)(3,4)(7,8), (1,4)(2,3)(5,8)(6,7)],
 [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]]
sage: lc = G.cosets(S, side='left'); lc
[[(), (1,3,5,7)(2,4,6,8), (1,5)(2,6)(3,7)(4,8), (1,7,5,3)(2,8,6,4)],
 [(2,8)(3,7)(4,6), (1,3)(4,8)(5,7), (1,5)(2,4)(6,8), (1,7)(2,6)(3,5)],
 [(1,2)(3,8)(4,7)(5,6), (1,4)(2,3)(5,8)(6,7), (1,6)(2,5)(3,4)(7,8), (1,8)(2,7)(3,6)(4,5)],
 [(1,2,3,4,5,6,7,8), (1,4,7,2,5,8,3,6), (1,6,3,8,5,2,7,4), (1,8,7,6,5,4,3,2)]]

sage: S.is_normal(G)
True
sage: rc == lc
False
sage: rc_sorted = [sorted(c) for c in rc]
sage: lc_sorted = [sorted(c) for c in lc]
sage: rc_sorted == lc_sorted
True

An example with the symmetry group of a regular tetrahedron and a subgroup that is not normal. Thus, the right and left cosets are different (and so are the representatives). With each individual coset sorted, a naive test of normality is possible.

sage: A = AlternatingGroup(4)
sage: face_turn = A.list()[4]; face_turn
(1,2,3)
sage: stabilizer = A.subgroup([face_turn])
sage: rc = A.cosets(stabilizer, side='right'); rc
[[(), (1,2,3), (1,3,2)],
 [(2,3,4), (1,3)(2,4), (1,4,2)],
 [(2,4,3), (1,4,3), (1,2)(3,4)],
 [(1,2,4), (1,4)(2,3), (1,3,4)]]
sage: lc = A.cosets(stabilizer, side='left'); lc
[[(), (1,2,3), (1,3,2)],
 [(2,3,4), (1,2)(3,4), (1,3,4)],
 [(2,4,3), (1,2,4), (1,3)(2,4)],
 [(1,4,2), (1,4,3), (1,4)(2,3)]]

sage: stabilizer.is_normal(A)
False
sage: rc_sorted = [sorted(c) for c in rc]
sage: lc_sorted = [sorted(c) for c in lc]
sage: rc_sorted == lc_sorted
False

TESTS:

The keyword side is checked for the two possible values.

sage: G = SymmetricGroup(3)
sage: S = G.subgroup([G("(1,2)")])
sage: G.cosets(S, side='junk')
Traceback (most recent call last):
...
ValueError: side should be 'right' or 'left', not junk

The subgroup argument is checked to see if it is a permutation group. Even a legitimate GAP object can be rejected.

sage: G=SymmetricGroup(3)
sage: G.cosets(gap(3))
Traceback (most recent call last):
...
TypeError: 3 is not a permutation group

The subgroup is verified as a subgroup of self.

sage: A = AlternatingGroup(3)
sage: G = SymmetricGroup(3)
sage: S = G.subgroup([G("(1,2)")])
sage: A.cosets(S)
Traceback (most recent call last):
...
ValueError: Subgroup of SymmetricGroup(3) generated by [(1,2)] is not a subgroup of AlternatingGroup(3)

AUTHOR:

  • Rob Beezer (2011-01-31)
degree()

Return the largest point moved by a permutation in this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: G.largest_moved_point()
4
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.largest_moved_point()
10
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.degree()
5

TODO: the name of this function is not good; this function should be deprecated in term of degree:

sage: P = PermutationGroup([[1,2,3,4]])
sage: P.largest_moved_point()
4
sage: P.cardinality()
1
derived_series()

Return the derived series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

sage: set_random_seed(0)
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.derived_series()  # random output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,4), (2,4)(3,5)]]
direct_product(other, maps=True)

Wraps GAP’s DirectProduct, Embedding, and Projection.

Sage calls GAP’s DirectProduct, which chooses an efficient representation for the direct product. The direct product of permutation groups will be a permutation group again. For a direct product D, the GAP operation Embedding(D,i) returns the homomorphism embedding the i-th factor into D. The GAP operation Projection(D,i) gives the projection of D onto the i-th factor. This method returns a 5-tuple: a permutation group and 4 morphisms.

INPUT:

  • self, other - permutation groups

OUTPUT:

  • D - a direct product of the inputs, returned as a permutation group as well
  • iota1 - an embedding of self into D
  • iota2 - an embedding of other into D
  • pr1 - the projection of D onto self (giving a splitting 1 - other - D - self - 1)
  • pr2 - the projection of D onto other (giving a splitting 1 - self - D - other - 1)

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: D = G.direct_product(G,False)
sage: D
Permutation Group with generators [(1,2,3,4), (5,6,7,8)]
sage: D,iota1,iota2,pr1,pr2 = G.direct_product(G)
sage: D; iota1; iota2; pr1; pr2
Permutation Group with generators [(1,2,3,4), (5,6,7,8)]
Homomorphism : Cyclic group of order 4 as a permutation group --> Permutation Group with generators [(1,2,3,4), (5,6,7,8)]
Homomorphism : Cyclic group of order 4 as a permutation group --> Permutation Group with generators [(1,2,3,4), (5,6,7,8)]
Homomorphism : Permutation Group with generators [(1,2,3,4), (5,6,7,8)] --> Cyclic group of order 4 as a permutation group
Homomorphism : Permutation Group with generators [(1,2,3,4), (5,6,7,8)] --> Cyclic group of order 4 as a permutation group
sage: g=D([(1,3),(2,4)]); g
(1,3)(2,4)
sage: d=D([(1,4,3,2),(5,7),(6,8)]); d
(1,4,3,2)(5,7)(6,8)
sage: iota1(g); iota2(g); pr1(d); pr2(d)
(1,3)(2,4)
(5,7)(6,8)
(1,4,3,2)
(1,3)(2,4)
exponent()

Computes the exponent of the group. The exponent e of a group G is the LCM of the orders of its elements, that is, e is the smallest integer such that g^e=1 for all g \in G.

EXAMPLES:

sage: G = AlternatingGroup(4)
sage: G.exponent()
6
gen(i=None)

Returns the i-th generator of self; that is, the i-th element of the list self.gens().

The argument i may be omitted if there is only one generator (but this will raise an error otherwise).

EXAMPLES:

We explicitly construct the alternating group on four elements:

sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(2,3,4), (1,2,3)]
sage: A4.gens()
[(2,3,4), (1,2,3)]
sage: A4.gen(0)
(2,3,4)
sage: A4.gen(1)
(1,2,3)
sage: A4.gens()[0]; A4.gens()[1]
(2,3,4)
(1,2,3)

sage: P1 = PermutationGroup([[(1,2)]]); P1.gen()
(1,2)
gens()

Return tuple of generators of this group. These need not be minimal, as they are the generators used in defining this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3)], [(1,2)]])
sage: G.gens()
[(1,2), (1,2,3)]

Note that the generators need not be minimal, though duplicates are removed:

sage: G = PermutationGroup([[(1,2)], [(1,3)], [(2,3)], [(1,2)]])
sage: G.gens()
[(2,3), (1,2), (1,3)]

We can use index notation to access the generators returned by self.gens:

sage: G = PermutationGroup([[(1,2,3,4), (5,6)], [(1,2)]])
sage: g = G.gens()
sage: g[0]
(1,2)
sage: g[1]
(1,2,3,4)(5,6)

TESTS:

We make sure that the trivial group gets handled correctly:

sage: SymmetricGroup(1).gens()
[()]
gens_small()

For this group, returns a generating set which has few elements. As neither irredundancy nor minimal length is proven, it is fast.

EXAMPLES:

sage: R = "(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24)" ## R = right
sage: U = "( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19)" ## U = top
sage: L = "( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35)" ## L = left
sage: F = "(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11)" ## F = front
sage: B = "(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27)" ## B = back or rear
sage: D = "(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40)" ## D = down or bottom
sage: G = PermutationGroup([R,L,U,F,B,D])
sage: len(G.gens_small())
2
group_id()

Return the ID code of this group, which is a list of two integers. Requires “optional” database_gap-4.4.x package.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.group_id()    # optional - database_gap
[12, 4]
has_element(item)

Returns boolean value of item in self - however ignores parentage.

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: gens = G.gens()
sage: H = DihedralGroup(4)
sage: g = G([(1,2,3,4)]); g
(1,2,3,4)
sage: G.has_element(g)
True
sage: h = H([(1,2),(3,4)]); h
(1,2)(3,4)
sage: G.has_element(h)
False
homology(n, p=0)

Computes the group homology H_n(G, F), where F = \ZZ if p=0 and F = \ZZ / p \ZZ if p > 0 is a prime. Wraps HAP’s GroupHomology function, written by Graham Ellis.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

AUTHORS:

  • David Joyner and Graham Ellis

The example below computes H_7(S_5, \ZZ), H_7(S_5, \ZZ / 2 \ZZ), H_7(S_5, \ZZ / 3 \ZZ), and H_7(S_5, \ZZ / 5 \ZZ), respectively. To compute the 2-part of H_7(S_5, \ZZ), use the homology_part function.

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.homology(7)                              # optional - gap_packages
Multiplicative Abelian Group isomorphic to C2 x C2 x C4 x C3 x C5
sage: G.homology(7,2)                              # optional - gap_packages
Multiplicative Abelian Group isomorphic to C2 x C2 x C2 x C2 x C2
sage: G.homology(7,3)                              # optional - gap_packages
Multiplicative Abelian Group isomorphic to C3
sage: G.homology(7,5)                              # optional - gap_packages
Multiplicative Abelian Group isomorphic to C5

REFERENCES:

homology_part(n, p=0)

Computes the p-part of the group homology H_n(G, F), where F = \ZZ if p=0 and F = \ZZ / p \ZZ if p > 0 is a prime. Wraps HAP’s Homology function, written by Graham Ellis, applied to the p-Sylow subgroup of G.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.homology_part(7,2)                              # optional - gap_packages
Multiplicative Abelian Group isomorphic to C2 x C2 x C2 x C2 x C4

AUTHORS:

  • David Joyner and Graham Ellis
id()

(Same as self.group_id().) Return the ID code of this group, which is a list of two integers. Requires “optional” database_gap-4.4.x package.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.group_id()    # optional - database_gap
[12, 4]
identity()

Return the identity element of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)]])
sage: e = G.identity()
sage: e
()
sage: g = G.gen(0)
sage: g*e
(1,2,3)(4,5)
sage: e*g
(1,2,3)(4,5)
intersection(other)

Returns the permutation group that is the intersection of self and other.

INPUT:

  • other - a permutation group.

OUTPUT:

A permutation group that is the set-theoretic intersection of self with other. The groups are viewed as subgroups of a symmetric group big enough to contain both group’s symbol sets. So there is no strict notion of the two groups being subgroups of a common parent.

EXAMPLES:

sage: H = DihedralGroup(4)

sage: K = CyclicPermutationGroup(4)
sage: H.intersection(K)
Permutation Group with generators [(1,2,3,4)]

sage: L = DihedralGroup(5)
sage: H.intersection(L)
Permutation Group with generators [(1,4)(2,3)]

sage: M = PermutationGroup(["()"])
sage: H.intersection(M)
Permutation Group with generators [()]

Some basic properties.

sage: H = DihedralGroup(4)
sage: L = DihedralGroup(5)
sage: H.intersection(L) == L.intersection(H)
True
sage: H.intersection(H) == H
True

The group other is verified as such.

sage: H = DihedralGroup(4)
sage: H.intersection('junk')
Traceback (most recent call last):
...
TypeError: junk is not a permutation group
irreducible_characters()

Returns a list of the irreducible characters of self.

EXAMPLES:

sage: irr = SymmetricGroup(3).irreducible_characters()
sage: [x.values() for x in irr]
[[1, -1, 1], [2, 0, -1], [1, 1, 1]]
is_abelian()

Return True if this group is abelian.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_abelian()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_abelian()
True
is_commutative()

Return True if this group is commutative.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_commutative()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_commutative()
True
is_cyclic()

Return True if this group is cyclic.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_cyclic()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_cyclic()
True
is_elementary_abelian()

Return True if this group is elementary abelian. An elementary abelian group is a finite abelian group, where every nontrivial element has order p, where p is a prime.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_elementary_abelian()
False
sage: G = PermutationGroup(['(1,2,3)','(4,5,6)'])
sage: G.is_elementary_abelian()
True
is_isomorphic(right)

Return True if the groups are isomorphic.

INPUT:

  • self - this group
  • right - a permutation group

OUTPUT:

  • boolean; True if self and right are isomorphic groups; False otherwise.

EXAMPLES:

sage: v = ['(1,2,3)(4,5)', '(1,2,3,4,5)']
sage: G = PermutationGroup(v)
sage: H = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_isomorphic(H)
False
sage: G.is_isomorphic(G)
True
sage: G.is_isomorphic(PermutationGroup(list(reversed(v))))
True
is_monomial()

Returns True if the group is monomial. A finite group is monomial if every irreducible complex character is induced from a linear character of a subgroup.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_monomial()
True
is_nilpotent()

Return True if this group is nilpotent.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_nilpotent()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_nilpotent()
True
is_normal(other)

Return True if this group is a normal subgroup of other.

EXAMPLES:

sage: AlternatingGroup(4).is_normal(SymmetricGroup(4))
True
sage: H = PermutationGroup(['(1,2,3)(4,5)'])
sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: H.is_normal(G)
False
is_perfect()

Return True if this group is perfect. A group is perfect if it equals its derived subgroup.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_perfect()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_perfect()
False
is_pgroup()

Returns True if this group is a p-group. A finite group is a p-group if its order is of the form p^n for a prime integer p and a nonnegative integer n.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3,4,5)'])
sage: G.is_pgroup()
True
is_polycyclic()

Return True if this group is polycyclic. A group is polycyclic if it has a subnormal series with cyclic factors. (For finite groups, this is the same as if the group is solvable - see is_solvable.)

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: G.is_polycyclic()
False
sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_polycyclic()
True
is_simple()

Returns True if the group is simple. A group is simple if it has no proper normal subgroups.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_simple()
False
is_solvable()

Returns True if the group is solvable.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_solvable()
True
is_subgroup(other)

Returns True if self is a subgroup of other.

EXAMPLES:

sage: G = AlternatingGroup(5)
sage: H = SymmetricGroup(5)
sage: G.is_subgroup(H)
True
is_supersolvable()

Returns True if the group is supersolvable. A finite group is supersolvable if it has a normal series with cyclic factors.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.is_supersolvable()
True
is_transitive()

Return True if self is a transitive group, i.e., if the action of self on [1..n] is transitive.

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.is_transitive()
True
sage: G = PermutationGroup(['(1,2)(3,4)(5,6)'])
sage: G.is_transitive()
False

Note that this differs from the definition in GAP, where IsTransitive returns whether the group is transitive on the set of points moved by the group.

sage: G = PermutationGroup([(2,3)])
sage: G.is_transitive()
False
sage: gap(G).IsTransitive()
true
isomorphism_to(right)

Return an isomorphism from self to right if the groups are isomorphic, otherwise None.

INPUT:

  • self - this group
  • right - a permutation group

OUTPUT:

  • None or a morphism of permutation groups.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: H = PermutationGroup(['(1,2,3)(4,5)'])
sage: G.isomorphism_to(H) is None
True
sage: G = PermutationGroup([(1,2,3), (2,3)])
sage: H = PermutationGroup([(1,2,4), (1,4)])
sage: G.isomorphism_to(H)
Homomorphism : Permutation Group with generators [(2,3), (1,2,3)] --> Permutation Group with generators [(1,2,4), (1,4)]
isomorphism_type_info_simple_group()

If the group is simple, then this returns the name of the group.

EXAMPLES:

sage: G = CyclicPermutationGroup(5)
sage: G.isomorphism_type_info_simple_group()
rec( series := "Z", parameter := 5, name := "Z(5)" )

TESTS: This shows that the issue at trac ticket 7360 is fixed:

sage: G = KleinFourGroup()
sage: G.is_simple()
False
sage: G.isomorphism_type_info_simple_group()
Traceback (most recent call last):
...
TypeError: Group must be simple.
largest_moved_point()

Return the largest point moved by a permutation in this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: G.largest_moved_point()
4
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.largest_moved_point()
10
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.degree()
5

TODO: the name of this function is not good; this function should be deprecated in term of degree:

sage: P = PermutationGroup([[1,2,3,4]])
sage: P.largest_moved_point()
4
sage: P.cardinality()
1
list(*args, **kwds)

Return list of all elements of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
sage: G.list()
[(), (3,4), (2,3), (2,3,4), (2,4,3), (2,4), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4), (1,2,4,3), (1,2,4), (1,3,2), (1,3,4,2), (1,3), (1,3,4), (1,3)(2,4), (1,3,2,4), (1,4,3,2), (1,4,2), (1,4,3), (1,4), (1,4,2,3), (1,4)(2,3)]
lower_central_series()

Return the lower central series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

sage: set_random_seed(0)
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.lower_central_series()  # random output
[Permutation Group with generators [(1,2,3)(4,5), (3,4)], Permutation Group with generators [(1,5)(3,4), (1,5)(2,3), (1,3)(2,4)]]
molien_series()

Returns the Molien series of a transitive permutation group. The function

M(x) = (1/|G|)\sum_{g\in G} \det(1-x*g)^{-1}

is sometimes called the “Molien series” of G. GAP’s MolienSeries is associated to a character of a group G. How are these related? A group G, given as a permutation group on n points, has a “natural” representation of dimension n, given by permutation matrices. The Molien series of G is the one associated to that permutation representation of G using the above formula. Character values then count fixed points of the corresponding permutations.

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.molien_series()     
1/(-x^15 + x^14 + x^13 - x^10 - x^9 - x^8 + x^7 + x^6 + x^5 - x^2 - x + 1)
sage: G = SymmetricGroup(3)
sage: G.molien_series()     
1/(-x^6 + x^5 + x^4 - x^2 - x + 1)
normal_subgroups()

Return the normal subgroups of this group as a (sorted in increasing order) list of permutation groups.

The normal subgroups of H = PSL(2,7) \times PSL(2,7) are 1, two copies of PSL(2,7) and H itself, as the following example shows.

EXAMPLES:

sage: G = PSL(2,7)
sage: D = G.direct_product(G)
sage: H = D[0]
sage: NH = H.normal_subgroups()
sage: len(NH)
4
sage: NH[1].is_isomorphic(G)
True
sage: NH[2].is_isomorphic(G)
True
normalizer(g)

Returns the normalizer of g in self.

EXAMPLES:

sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4)]])
sage: g = G([(1,3)])
sage: G.normalizer(g)
Permutation Group with generators [(2,4), (1,3)]
sage: g = G([(1,2,3,4)])
sage: G.normalizer(g)
Permutation Group with generators [(2,4), (1,2,3,4), (1,3)(2,4)]
sage: H = G.subgroup([G([(1,2,3,4)])])
sage: G.normalizer(H)
Permutation Group with generators [(2,4), (1,2,3,4), (1,3)(2,4)]
normalizes(other)

Returns True if the group other is normalized by self. Wraps GAP’s IsNormal function.

A group G normalizes a group U if and only if for every g \in G and u \in U the element u^g is a member of U. Note that U need not be a subgroup of G.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)(4,5)'])
sage: H = PermutationGroup(['(1,2,3)(4,5)', '(1,2,3,4,5)'])
sage: H.normalizes(G)
False
sage: G = SymmetricGroup(3)
sage: H = PermutationGroup( [ (4,5,6) ] )
sage: G.normalizes(H)
True
sage: H.normalizes(G)
True

In the last example, G and H are disjoint, so each normalizes the other.

orbit(*args, **kwds)

Return the orbit of the given integer under the group action.

EXAMPLES:

sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
sage: G.orbit(3)
[3, 4, 1]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.orbit(3)                                                 
[3, 4, 10, 1, 2]
orbits(*args, **kwds)

Returns the orbits of [1,2,...,degree] under the group action.

EXAMPLES:

sage: G = PermutationGroup([ [(3,4)], [(1,3)] ]) 
sage: G.orbits()
[[1, 3, 4], [2]]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.orbits()
[[1, 2, 3, 4, 10], [5], [6], [7], [8], [9]]

The answer is cached:

sage: G.orbits() is G.orbits()
True           

AUTHORS:

  • Nathan Dunfield
order()

Return the number of elements of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.order()
12
sage: G = PermutationGroup([()])
sage: G.order()
1
sage: G = PermutationGroup([])
sage: G.order()
1
poincare_series(p=2, n=10)

Returns the Poincare series of G \mod p (p \geq 2 must be a prime), for n large. In other words, if you input a finite group G, a prime p, and a positive integer n, it returns a quotient of polynomials f(x) = P(x) / Q(x) whose coefficient of x^k equals the rank of the vector space H_k(G, \ZZ / p \ZZ), for all k in the range 1 \leq k \leq n.

REQUIRES: GAP package HAP (in gap_packages-*.spkg).

EXAMPLES:

sage: G = SymmetricGroup(5)
sage: G.poincare_series(2,10)                              # optional - gap_packages
(x^2 + 1)/(x^4 - x^3 - x + 1)
sage: G = SymmetricGroup(3)
sage: G.poincare_series(2,10)                              # optional - gap_packages
1/(-x + 1)

AUTHORS:

  • David Joyner and Graham Ellis
quotient(N)

Returns the quotient of this permutation group by the normal subgroup N.

Wraps the GAP operator “/”.

EXAMPLES:

sage: G = PermutationGroup([(1,2,3), (2,3)])
sage: N = PermutationGroup([(1,2,3)])
sage: G.quotient(N)
Permutation Group with generators [(1,2)]
quotient_group(N)

This function has been deprecated and will be removed in a future version of Sage; use quotient instead.

Original docstring follows.

Returns the quotient of this permutation group by the normal subgroup N.

Wraps the GAP operator “/”.

TESTS:

sage: G = PermutationGroup([(1,2,3), (2,3)])
sage: N = PermutationGroup([(1,2,3)])
sage: G.quotient_group(N)
doctest:...: DeprecationWarning: quotient_group() is deprecated; use quotient() instead.
Permutation Group with generators [(1,2)]
random_element()

Return a random element of this group.

EXAMPLES:

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: a = G.random_element()
sage: a in G
True
sage: a.parent() is G
True
sage: a^6
()
smallest_moved_point(*args, **kwds)

Return the smallest point moved by a permutation in this group.

EXAMPLES:

sage: G = PermutationGroup([[(3,4)], [(2,3,4)]])
sage: G.smallest_moved_point()
2
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.smallest_moved_point()
1
stabilizer(position)

Return the subgroup of self which stabilize the given position. self and its stabilizers must have same degree.

EXAMPLES:

sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])
sage: G.stabilizer(1)
Permutation Group with generators [(3,4)]
sage: G.stabilizer(3)
Permutation Group with generators [(1,4)]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.stabilizer(10)
Permutation Group with generators [(1,2)(3,4), (2,3,4)]
sage: G.stabilizer(1) 
Permutation Group with generators [(2,3)(4,10), (2,10,4)]
sage: G = PermutationGroup([[(2,3,4)],[(6,7)]])
sage: G.stabilizer(1)
Permutation Group with generators [(6,7), (2,3,4)]
sage: G.stabilizer(2)
Permutation Group with generators [(6,7)]
sage: G.stabilizer(3)
Permutation Group with generators [(6,7)]
sage: G.stabilizer(4)
Permutation Group with generators [(6,7)]
sage: G.stabilizer(5)
Permutation Group with generators [(6,7), (2,3,4)]
sage: G.stabilizer(6)
Permutation Group with generators [(2,3,4)]
sage: G.stabilizer(7)
Permutation Group with generators [(2,3,4)]
sage: G.stabilizer(8)
Permutation Group with generators [(6,7), (2,3,4)]
strong_generating_system(base_of_group=None)

Return a Strong Generating System of self according the given base for the right action of self on itself.

base_of_group is a list of the positions on which self acts, in any order. The algorithm returns a list of transversals and each transversal is a list of permutations. By default, base_of_group is [1, 2, 3, ..., d] where d is the degree of the group.

For base_of_group = [ \mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_d] let G_i be the subgroup of G = self which stabilizes \mathrm{pos}_1, \mathrm{pos}_2, \dots , \mathrm{pos}_i, so

G = G_0 \supset G_1 \supset G_2 \supset \dots \supset G_n = \{e\}

Then the algorithm returns [ G_i.\mathrm{transversals}(\mathrm{pos}_{i+1})]_{1 \leq i \leq n}

INPUT:

  • base_of_group (optional) – default: [1, 2, 3, ..., d] – a list containing the integers 1, 2, \dots , d in any order (d is the degree of self)

OUTPUT:

  • A list of lists of permutations from the group, which form a strong generating system.

TESTS:

sage: G = SymmetricGroup(10)
sage: H = PermutationGroup([G.random_element() for i in range(randrange(1,3,1))])
sage: prod(map(lambda x : len(x), H.strong_generating_system()),1) == H.cardinality()
True

EXAMPLES:

sage: G = PermutationGroup([[(7,8)],[(3,4)],[(4,5)]])
sage: G.strong_generating_system()
[[()], [()], [(), (3,4,5), (3,5)], [(), (4,5)], [()], [()], [(), (7,8)], [()]]
sage: G = PermutationGroup([[(1,2,3,4)],[(1,2)]])
sage: G.strong_generating_system()
[[(), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3)], [(), (2,3,4), (2,4,3)], [(), (3,4)], [()]]
sage: G = PermutationGroup([[(1,2,3)],[(4,5,7)],[(1,4,6)]])
sage: G.strong_generating_system()                         
[[(), (1,2,3), (1,4,6), (1,3,2), (1,5,7,4,6), (1,6,4), (1,7,5,4,6)], [(), (2,6,3), (2,3,6), (2,5,6,3)(4,7), (2,7,5,6,3), (2,4,5,6,3)], [(), (3,6)(5,7), (3,5,6), (3,7,4,5,6), (3,4,7,5,6)], [(), (4,5)(6,7), (4,7)(5,6), (4,6)(5,7)], [(), (5,6,7), (5,7,6)], [()], [()]]
sage: G = PermutationGroup([[(1,2,3)],[(2,3,4)],[(3,4,5)]])
sage: G.strong_generating_system([5,4,3,2,1])              
[[(), (1,5,3,4,2), (1,5,4,3,2), (1,5)(2,3), (1,5,2)], [(), (1,3)(2,4), (1,2)(3,4), (1,4)(2,3)], [(), (1,3,2), (1,2,3)], [()], [()]]
sage: G = PermutationGroup([[(3,4)]])
sage: G.strong_generating_system()
[[()], [()], [(), (3,4)], [()]]
sage: G.strong_generating_system(base_of_group=[3,1,2,4])
[[(), (3,4)], [()], [()], [()]]
sage: G = TransitiveGroup(12,17)                # optional
sage: G.strong_generating_system()              # optional
[[(), (1,4,11,2)(3,6,5,8)(7,10,9,12), (1,8,3,2)(4,11,10,9)(5,12,7,6), (1,7)(2,8)(3,9)(4,10)(5,11)(6,12), (1,12,7,2)(3,10,9,8)(4,11,6,5), (1,11)(2,8)(3,5)(4,10)(6,12)(7,9), (1,10,11,8)(2,3,12,5)(4,9,6,7), (1,3)(2,8)(4,10)(5,7)(6,12)(9,11), (1,2,3,8)(4,9,10,11)(5,6,7,12), (1,6,7,8)(2,3,4,9)(5,10,11,12), (1,5,9)(3,11,7), (1,9,5)(3,7,11)], [(), (2,6,10)(4,12,8), (2,10,6)(4,8,12)], [()], [()], [()], [()], [()], [()], [()], [()], [()], [()]]
subgroup(gens)

Wraps the PermutationGroup_subgroup constructor. The argument gens is a list of elements of self.

EXAMPLES:

sage: G = PermutationGroup([(1,2,3),(3,4,5)])
sage: g = G((1,2,3))
sage: G.subgroup([g])
Subgroup of Permutation Group with generators [(3,4,5), (1,2,3)] generated by [(1,2,3)]
subgroups()

Returns a list of all the subgroups of self.

OUTPUT:

Each possible subgroup of self is contained once in the returned list. The list is in order, according to the size of the subgroups, from the trivial subgroup with one element on through up to the whole group. Conjugacy classes of subgroups are contiguous in the list.

Warning

For even relatively small groups this method can take a very long time to execute, or create vast amounts of output. Likely both. Its purpose is instructional, as it can be useful for studying small groups. The 156 subgroups of the full symmetric group on 5 symbols of order 120, S_5, can be computed in about a minute on commodity hardware in 2011. The 64 subgroups of the cyclic group of order 30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13 takes about twice as long.

For faster results, which still exhibit the structure of the possible subgroups, use conjugacy_classes_subgroups().

EXAMPLES:

sage: G = SymmetricGroup(3)
sage: G.subgroups()
[Permutation Group with generators [()],
 Permutation Group with generators [(2,3)],
 Permutation Group with generators [(1,2)],
 Permutation Group with generators [(1,3)],
 Permutation Group with generators [(1,2,3)],
 Permutation Group with generators [(1,3,2), (1,2)]]

sage: G = CyclicPermutationGroup(14)
sage: G.subgroups()
[Permutation Group with generators [()],
 Permutation Group with generators [(1,8)(2,9)(3,10)(4,11)(5,12)(6,13)(7,14)],
 Permutation Group with generators [(1,3,5,7,9,11,13)(2,4,6,8,10,12,14)],
 Permutation Group with generators [(1,2,3,4,5,6,7,8,9,10,11,12,13,14)]]

AUTHOR:

  • Rob Beezer (2011-01-24)
sylow_subgroup(p)

Returns a Sylow p-subgroup of the finite group G, where p is a prime. This is a p-subgroup of G whose index in G is coprime to p. Wraps the GAP function SylowSubgroup.

EXAMPLES:

sage: G = PermutationGroup(['(1,2,3)', '(2,3)'])
sage: G.sylow_subgroup(2)
Permutation Group with generators [(2,3)]
sage: G.sylow_subgroup(5)
Permutation Group with generators [()]
transversals(integer)

If G is a permutation group acting on the set X = \{1, 2, ...., n\} and H is the stabilizer subgroup of <integer>, a right (respectively left) transversal is a set containing exactly one element from each right (respectively left) coset of H. This method returns a right transversal of self by the stabilizer of self on <integer> position.

EXAMPLES:

sage: G = PermutationGroup([ [(3,4)], [(1,3)] ])           
sage: G.transversals(1)
[(), (1,3,4), (1,4,3)]
sage: G = PermutationGroup([[(1,2),(3,4)], [(1,2,3,4,10)]])
sage: G.transversals(1)                                    
[(), (1,2)(3,4), (1,3,2,10,4), (1,4,2,10,3), (1,10,4,3,2)]
trivial_character()

Returns the trivial character of self.

EXAMPLES:

sage: SymmetricGroup(3).trivial_character()
Character of Symmetric group of order 3! as a permutation group
upper_central_series()

Return the upper central series of this group as a list of permutation groups.

EXAMPLES:

These computations use pseudo-random numbers, so we set the seed for reproducible testing.

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G.upper_central_series()
[Permutation Group with generators [()]]
class sage.groups.perm_gps.permgroup.PermutationGroup_subgroup(ambient, gens, from_group=False, check=True, canonicalize=True)

Bases: sage.groups.perm_gps.permgroup.PermutationGroup_generic

Subgroup subclass of PermutationGroup_generic, so instance methods are inherited.

EXAMPLES:

sage: G = CyclicPermutationGroup(4)
sage: gens = G.gens()
sage: H = DihedralGroup(4)
sage: PermutationGroup_subgroup(H,list(gens))
Subgroup of Dihedral group of order 8 as a permutation group generated by [(1,2,3,4)]
sage: K=PermutationGroup_subgroup(H,list(gens))
sage: K.list()
[(), (1,2,3,4), (1,3)(2,4), (1,4,3,2)]
sage: K.ambient_group()
Dihedral group of order 8 as a permutation group
sage: K.gens()
[(1,2,3,4)]
ambient_group()

Return the ambient group related to self.

EXAMPLES:

An example involving the dihedral group on four elements, D_8:

sage: G = DihedralGroup(4)
sage: H = CyclicPermutationGroup(4)
sage: gens = H.gens()
sage: S = PermutationGroup_subgroup(G, list(gens))
sage: S.ambient_group()
Dihedral group of order 8 as a permutation group
sage: S.ambient_group() == G
True
gens()

Return the generators for this subgroup.

EXAMPLES:

An example involving the dihedral group on four elements, D_8:

sage: G = DihedralGroup(4)
sage: H = CyclicPermutationGroup(4)
sage: gens = H.gens()
sage: S = PermutationGroup_subgroup(G, list(gens))
sage: S.gens()
[(1,2,3,4)]
sage: S.gens() == list(H.gens())
True
sage.groups.perm_gps.permgroup.direct_product_permgroups(P)

Takes the direct product of the permutation groups listed in P.

EXAMPLES:

sage: G1 = AlternatingGroup([1,2,4,5])
sage: G2 = AlternatingGroup([3,4,6,7])
sage: D = direct_product_permgroups([G1,G2,G1])
sage: D.order()
1728
sage: D = direct_product_permgroups([G1])
sage: D==G1
True
sage: direct_product_permgroups([])
Symmetric group of order 0! as a permutation group
sage.groups.perm_gps.permgroup.from_gap_list(G, src)

Convert a string giving a list of GAP permutations into a list of elements of G.

EXAMPLES:

sage: from sage.groups.perm_gps.permgroup import from_gap_list
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: L = from_gap_list(G, "[(1,2,3)(4,5), (3,4)]"); L
[(1,2,3)(4,5), (3,4)]
sage: L[0].parent() is G
True
sage: L[1].parent() is G
True
sage.groups.perm_gps.permgroup.load_hap()

Load the GAP hap package into the default GAP interpreter interface. If this fails, try one more time to load it.

EXAMPLES:

sage: sage.groups.perm_gps.permgroup.load_hap()

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