22.6 Permutation groups

Module: sage.groups.perm_gps.permgroup

Permutation groups

A permutation group is a finite group G whose elements are permutations of a given finite set X (i.e., bijections X -> 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.

(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.

Author: - 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) - Stein and 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

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.

Module-level Functions

PermutationGroup( x, [from_group=False], [check=True])

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

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

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          # requires optional database_gap
Transitive group number 3 of degree 4
sage: H.gens()                            # requires 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( [ (1,2,3)(4,5), (3,4) ] ), Group( [ (1,5)(3,4), (1,5)(2,3),
(1,5,4) ] ) ]

direct_product_permgroups( P)

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

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 1! as a permutation group

from_gap_list( G, src)
Convert a string giving a list of GAP permutations into a list of elements of G.

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

gap_format( x)

Put a permutation in Gap format, as a string.

Class: PermutationGroup_generic

class PermutationGroup_generic

sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G
Permutation Group with generators [(1,2,3)(4,5), (3,4)]
sage: G.center()
Permutation Group with generators [()]
sage: G.group_id()          # requires optional database_gap
[120, 34]
sage: n = G.order(); n
120
sage: G = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: loads(G.dumps()) == G
True
PermutationGroup_generic( self, gens, [from_group=False], [check=True])

Initializes instance of self, a PermutationGroup_generic object. The flag from_group determines whether or not self should be initialized as a Gap Group object and sent to the Gap interpreter. If gens are Gap objects, then self is initialized as a Gap Group.

We explicitly construct the alternating group on four elements.

sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(1,2,3), (2,3,4)]
sage: A4.__init__([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(1,2,3), (2,3,4)]
sage: A4.center()
Permutation Group with generators [()]
sage: loads(A4.dumps()) == A4
True

Functions: cayley_table,$ \,$ center,$ \,$ character_table,$ \,$ cohomology,$ \,$ cohomology_part,$ \,$ composition_series,$ \,$ conjugacy_classes_representatives,$ \,$ conjugacy_classes_subgroups,$ \,$ degree,$ \,$ derived_series,$ \,$ direct_product,$ \,$ exponent,$ \,$ gen,$ \,$ gens,$ \,$ gens_small,$ \,$ group_id,$ \,$ has_element,$ \,$ homology,$ \,$ homology_part,$ \,$ id,$ \,$ identity,$ \,$ is_abelian,$ \,$ is_commutative,$ \,$ is_cyclic,$ \,$ is_elementary_abelian,$ \,$ is_isomorphic,$ \,$ is_monomial,$ \,$ is_nilpotent,$ \,$ is_normal,$ \,$ is_perfect,$ \,$ is_pgroup,$ \,$ is_polycyclic,$ \,$ is_simple,$ \,$ is_solvable,$ \,$ is_subgroup,$ \,$ is_supersolvable,$ \,$ is_transitive,$ \,$ isomorphism_to,$ \,$ isomorphism_type_info_simple_group,$ \,$ largest_moved_point,$ \,$ list,$ \,$ lower_central_series,$ \,$ molien_series,$ \,$ multiplication_table,$ \,$ normal_subgroups,$ \,$ normalizer,$ \,$ normalizes,$ \,$ orbits,$ \,$ order,$ \,$ poincare_series,$ \,$ quotient_group,$ \,$ random_element,$ \,$ smallest_moved_point,$ \,$ subgroup,$ \,$ sylow_subgroup,$ \,$ upper_central_series

cayley_table( self, [names=x])

Returns the multiplication table, or Cayley table, of the finite group G in the form of a matrix with symbolic coefficients. This function is useful for learning, teaching, and exploring elementary group theory. Of course, G must be a group of low order.

As the last line below illustrates, the ordering used here in the first row is the same as in G.list().

sage: G = PermutationGroup(['(1,2,3)', '(2,3)'])
sage: G.cayley_table()
[x0 x1 x2 x3 x4 x5]
[x1 x0 x3 x2 x5 x4]
[x2 x4 x0 x5 x1 x3]
[x3 x5 x1 x4 x0 x2]
[x4 x2 x5 x0 x3 x1]
[x5 x3 x4 x1 x2 x0]
sage: G.list()[3]*G.list()[3] == G.list()[4]
True
sage: G.cayley_table("y")
[y0 y1 y2 y3 y4 y5]
[y1 y0 y3 y2 y5 y4]
[y2 y4 y0 y5 y1 y3]
[y3 y5 y1 y4 y0 y2]
[y4 y2 y5 y0 y3 y1]
[y5 y3 y4 y1 y2 y0]
sage: G.cayley_table(names="abcdef")
[a b c d e f]
[b a d c f e]
[c e a f b d]
[d f b e a c]
[e c f a d b]
[f d e b c a]

center( self)

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

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 [()]

character_table( self)

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

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(

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(

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, ..., 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, ...,v_n]$ = character_table_values(G) $ ^{-1}[c_1, ...,c_n]$ .

Author: - David Joyner and William Stein (2006-01-04)

cohomology( self, n, [p=0])

Computes the group cohomology H_n(G, F), where F = Z if p=0 and F = Z/pZ if p >0 is a prime. Wraps HAP's GroupHomology function, written by Graham Ellis.

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

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

This computes $ H^4(S_3,ZZ)$ , $ H^4(S_3,ZZ/2ZZ)$ , resp.

Author: David Joyner and Graham Ellis

REFERENCES: G. Ellis, "Computing group resolutions", J. Symbolic Computation. Vol.38, (2004)1077-1118 (Available at http://hamilton.nuigalway.ie/. D. Joyner, "A primer on computational group homology and cohomology", http://front.math.ucdavis.edu/0706.0549

cohomology_part( self, n, [p=0])

Computes the p-part of the group cohomology $ H^n(G, F)$ , where $ F = Z$ if $ p=0$ and $ F = Z/pZ$ 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).

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

Author: David Joyner and Graham Ellis

composition_series( self)

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

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 [()]]

conjugacy_classes_representatives( self)

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

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)]

Author: David Joyner and William Stein (2006-01-04)

conjugacy_classes_subgroups( self)

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

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)]]

Author: David Joyner (2006-10)

degree( self)

Synonym for largest_moved_point().

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

derived_series( self)

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

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( self, 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.

Input:

self, other
- permutation groups

This method returns a 5-tuple - a permutation groups and 4 morphisms.

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)

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( self)

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$ .

sage: G = AlternatingGroup(4)
sage: G.exponent()
6

gen( self, i)

Returns the ith generator of self; that is, the ith element of the list self.gens().

We explicitly construct the alternating group on four elements:

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

gens( self)

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

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

Note that the generators need not be minimal.

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

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

gens_small( self)

Returns a generating set of G which has few elements. As neither irredundancy, nor minimal length is proven, it is fast.

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( self)

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

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.group_id()    # requires optional database_gap-4.4.6 package
[12, 4]

has_element( self, item)

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

       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( self, n, [p=0])

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

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

Author: David Joyner and Graham Ellis

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

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

REFERENCES: G. Ellis, "Computing group resolutions", J. Symbolic Computation. Vol.38, (2004)1077-1118 (Available at http://hamilton.nuigalway.ie/. D. Joyner, "A primer on computational group homology and cohomology", http://front.math.ucdavis.edu/0706.0549

homology_part( self, n, [p=0])

Computes the $ p$ -part of the group homology $ H_n(G, F)$ , where $ F = Z$ if $ p=0$ and $ F = Z/pZ$ 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).

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

Author: David Joyner and Graham Ellis

id( self)

(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.

sage: G = PermutationGroup([[(1,2,3),(4,5)], [(1,2)]])
sage: G.group_id()    # requires optional database_gap-4.4.6 package
[12, 4]

identity( self)

Return the identity element of this group.

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)

is_abelian( self)

Return True if this group is abelian.

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( self)

Return True if this group is commutative.

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( self)

Return True if this group is cyclic.

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( self)

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.

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( self, right)

Return True if the groups are isomorphic. If mode="verbose" then an isomorphism is printed.

Input:

self
- this group
right
- a permutation group
Output: bool

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( self)

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.

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

is_nilpotent( self)

Return True if this group is nilpotent.

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( self, other)

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

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( self)

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

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( self)

Returns True if the 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.

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

is_polycyclic( self)

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)].)

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( self)

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

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

is_solvable( self)

Returns True if the group is solvable.

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

is_subgroup( self, other)

Returns true if self is a subgroup of other.

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

is_supersolvable( self)

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

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

is_transitive( self)

Return True if self is a transitive group, i.e. if the action of self on self.set() is transitive.

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

isomorphism_to( self, right)

Return an isomorphism 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.

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 [(1,2,3), (2,3)] -->
Permutation Group with generators [(1,2,4), (1,4)]

isomorphism_type_info_simple_group( self)

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

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

largest_moved_point( self)

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

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

list( self)

Return list of all elements of this group.

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( self)

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

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( self)

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

$\displaystyle M(x) = (1/\vert G\vert)\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.

sage: G = SymmetricGroup(5)
sage: G.molien_series()                              # requires optional gap_packages
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()                              # requires optional gap_packages
1/(-x^6 + x^5 + x^4 - x^2 - x + 1)

multiplication_table( self, [names=x])

Returns the multiplication table, or Cayley table, of the finite group G in the form of a matrix with symbolic coefficients. This function is useful for learning, teaching, and exploring elementary group theory. Of course, G must be a group of low order.

As the last line below illustrates, the ordering used here in the first row is the same as in G.list().

sage: G = PermutationGroup(['(1,2,3)', '(2,3)'])
sage: G.cayley_table()
[x0 x1 x2 x3 x4 x5]
[x1 x0 x3 x2 x5 x4]
[x2 x4 x0 x5 x1 x3]
[x3 x5 x1 x4 x0 x2]
[x4 x2 x5 x0 x3 x1]
[x5 x3 x4 x1 x2 x0]
sage: G.list()[3]*G.list()[3] == G.list()[4]
True
sage: G.cayley_table("y")
[y0 y1 y2 y3 y4 y5]
[y1 y0 y3 y2 y5 y4]
[y2 y4 y0 y5 y1 y3]
[y3 y5 y1 y4 y0 y2]
[y4 y2 y5 y0 y3 y1]
[y5 y3 y4 y1 y2 y0]
sage: G.cayley_table(names="abcdef")
[a b c d e f]
[b a d c f e]
[c e a f b d]
[d f b e a c]
[e c f a d b]
[f d e b c a]

normal_subgroups( self)

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)xPSL(2,7)$ are $ 1$ , two copies of $ PSL(2,7)$ and $ H$ itself, as the following example shows.

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( self, g)

Returns the normalizer of g in self.

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

normalizes( self, other)

Returns True if the group other is normalized by the 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.

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.

orbits( self)

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

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

Author: Nathan Dunfield

order( self)

Return the number of elements of this group.

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

poincare_series( self, [p=2], [n=10])

Returns the Poincare series of G mod p (p must be a prime), for n>1 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/pZZ)$ , for all k in the range $ 1\leq k \leq n$ .

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

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

Author: David Joyner and Graham Ellis

quotient_group( self, N)

Returns the quotient group permgp/N, where N is a normal subgroup. Wraps the GAP operator "/".

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

random_element( self)

Return a random element of this group.

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

smallest_moved_point( self)

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

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

subgroup( self, gens)

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

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 [(1,2,3), (3,4,5)] generated
by [(1,2,3)]

sylow_subgroup( self, p)

Returns a Sylow p-subgroups 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.

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 [()]

upper_central_series( self)

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

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.upper_central_series()
[Permutation Group with generators [()]]

Special Functions: __call__,$ \,$ __cmp__,$ \,$ __contains__,$ \,$ __init__,$ \,$ __iter__,$ \,$ _coerce_impl,$ \,$ _gap_init_,$ \,$ _latex_,$ \,$ _magma_init_,$ \,$ _repr_

__call__( self, x, [check=True])

Coerce x into this permutation group.

The input can be either a string that defines a permutation in cycle notation, a permutation group element, a list of integers that gives the permutation as a mapping, a list of tuples, or the integer 1.

We illustrate each way to make a permutation in S4:

sage: G = SymmetricGroup(4)
sage: G((1,2,3,4))
(1,2,3,4)
sage: G([(1,2),(3,4)])
(1,2)(3,4)
sage: G('(1,2)(3,4)')
(1,2)(3,4)
sage: G('(1,2)(3)(4)')
(1,2)
sage: G(((1,2,3),(4,)))
(1,2,3)
sage: G(((1,2,3,4),))
(1,2,3,4)
sage: G([1,2,4,3])
(3,4)
sage: G([2,3,4,1])
(1,2,3,4)
sage: G(G((1,2,3,4)))
(1,2,3,4)
sage: G(1)
()

Some more examples:

sage: G = PermutationGroup([(1,2,3,4)])
sage: G([(1,3), (2,4)])
(1,3)(2,4)
sage: G(G.0^3)
(1,4,3,2)
sage: G(1)
()
sage: G((1,4,3,2))
(1,4,3,2)
sage: G([(1,2)])
Traceback (most recent call last):
...
TypeError: permutation (1,2) not in Permutation Group with generators
[(1,2,3,4)]

__cmp__( self, right)

Compare self and right.

The ordering is whatever it is in Gap.

sage: G1 = PermutationGroup([[(1,2,3),(4,5)],[(3,4)]])
sage: G2 = PermutationGroup([[(1,2,3),(4,5)]])
sage: G1 > G2
True
sage: G1 < G2
False

__contains__( self, item)

Returns boolean value of "item in self"

       sage: G = SymmetricGroup(16)
sage: g = G.gen(0)
sage: h = G.gen(1)
sage: g^7*h*g*h in G
True
sage: G = SymmetricGroup(4)
sage: g = G((1,2,3,4))
sage: h = G((1,2))
       sage: H = PermutationGroup([[(1,2,3,4)], [(1,2),(3,4)]])
sage: g in H
True
sage: h in H
False

__iter__( self)

Return an iterator over the elements of this group.

sage: G = PermutationGroup([[(1,2,3)], [(1,2)]])
sage: [a for a in G]
[(), (2,3), (1,2), (1,2,3), (1,3,2), (1,3)]

_coerce_impl( self, x)

Implicit coercion of x into self.

We illustrate some arithmetic that involves implicit coercion of elements in different permutation groups.

sage: g1 = PermutationGroupElement([(1,2),(3,4,5)])
sage: g1.parent()
Symmetric group of order 5! as a permutation group
sage: g2 = PermutationGroupElement([(1,2)])
sage: g2.parent()
Symmetric group of order 2! as a permutation group
sage: g1*g2
(3,4,5)
sage: g2*g2
()
sage: g2*g1
(3,4,5)

We try to implicitly coerce in a non-permutation, which raises a TypeError:

sage: G = PermutationGroup([[(1,2,3,4)], [(1,2)]])
sage: G._coerce_impl(1)
Traceback (most recent call last):
...
TypeError: no implicit coercion of element into permutation group

_gap_init_( self)

Returns a string showing how to declare / initialize self in Gap. Stored in the self.__gap attribute.

The _gap_init_ method shows how you would define the Sage PermutationGroup_generic object in Gap:

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

_latex_( self)

Method for describing self in LaTeX. Encapsulates self.gens() in angle brackets to denote that self in generated by these elements. Called by the latex() function.

We explicitly construct the alternating group on four elements.

sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(1,2,3), (2,3,4)]
sage: latex(A4)
\langle (1,2,3), (2,3,4) \rangle
sage: A4._latex_()
'\\langle (1,2,3), (2,3,4) \\rangle'

_magma_init_( self)

Returns a string showing how to declare / intialize self in Magma.

We explicitly construct the alternating group on four elements. In Magma, one would type the string below to construct the group.

sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(1,2,3), (2,3,4)]
sage: A4._magma_init_()
'PermutationGroup<4 | (1,2,3), (2,3,4)>'

_repr_( self)

Returns a string describing self.

We explicitly construct the alternating group on four elements. Note that the AlternatingGroup class has its own representation string:

sage: A4 = PermutationGroup([[(1,2,3)],[(2,3,4)]]); A4
Permutation Group with generators [(1,2,3), (2,3,4)]
sage: A4._repr_()
'Permutation Group with generators [(1,2,3), (2,3,4)]'
sage: AlternatingGroup(4)._repr_()
'Alternating group of order 4!/2 as a permutation group'

Class: PermutationGroup_subgroup

class PermutationGroup_subgroup
Subgroup subclass of PermutationGroup_generic, so instance methods are inherited.

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)]
PermutationGroup_subgroup( self, ambient, gens, [from_group=False], [check=True])

Initialization method for the PermutationGroup_subgroup class.

INPUTS: ambient - the ambient group from which to construct this subgroup gens - the generators of the subgroup from_group - True: subroup is generated from a Gap string representation of the generators check- True: checks if gens are indeed elements of the ambient group

An example involving the dihedral group on four elements. $ D_8$ contains a cyclic subgroup or order four:

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

However, $ D_8$ does not contain a cyclic subgroup of order three:

sage: G = DihedralGroup(4)
sage: H = CyclicPermutationGroup(3)
sage: gens = H.gens()
sage: S = PermutationGroup_subgroup(G,list(gens))
Traceback (most recent call last):
...
TypeError: each generator must be in the ambient group

Functions: ambient_group,$ \,$ gens

ambient_group( self)

Return the ambient group related to self.

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( self)

Return the generators for this subgroup.

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

Special Functions: __cmp__,$ \,$ __init__,$ \,$ _latex_,$ \,$ _repr_

__cmp__( self, other)

Compare self and other. If self and other are in a common ambient group, then self <= other precisely if self is contained in other.

       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: G<K
       False
       sage: G>K
       False

_latex_( self)

Return latex representation of this group.

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: latex(S)
Subgroup of Dihedral group of order 8 as a permutation group generated by
[(1,2,3,4)]
sage: S._latex_()
'Subgroup of Dihedral group of order 8 as a permutation group generated by
[(1,2,3,4)]'

_repr_( self)

Returns a string representation / description of the permutation subgroup.

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
Subgroup of Dihedral group of order 8 as a permutation group generated by
[(1,2,3,4)]
sage: S._repr_()
'Subgroup of Dihedral group of order 8 as a permutation group generated by
[(1,2,3,4)]'

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