“Named” Permutation groups (such as the symmetric group, S_n)¶

You can construct the following permutation groups:

– SymmetricGroup, $S_n$ of order $n!$ (n can also be a list $X$ of distinct: positive integers, in which case it returns $S_X$)
– AlternatingGroup, $A_n$ or order $n!/2$ (n can also be a list $X$: of distinct positive integers, in which case it returns $A_X$)

– DihedralGroup, $D_n$ of order $2n$

– CyclicPermutationGroup, $C_n$ of order $n$

– DiCyclicGroup, nonabelian groups of order $4m$ with a unique element of order 2

– TransitiveGroup, $n^{th}$ transitive group of degree $d$: from the GAP tables of transitive groups (requires the “optional” package database_gap)

– TransitiveGroups(d), TransitiveGroups(), set of all of the above

– MathieuGroup(degree), Mathieu group of degree 9, 10, 11, 12, 21, 22, 23, or 24.

– KleinFourGroup, subgroup of $S_4$ of order $4$ which is not $C_2 \times C_2$

– QuaternionGroup, non-abelian group of order $8$ , $\{\pm 1, \pm I, \pm J, \pm K\}$

– PGL(n,q), projective general linear group of $n\times n$ matrices over: the finite field GF(q)
– PSL(n,q), projective special linear group of $n\times n$ matrices over: the finite field GF(q)
– PSp(2n,q), projective symplectic linear group of $2n\times 2n$ matrices: over the finite field GF(q)
– PSU(n,q), projective special unitary group of $n\times n$ matrices having: coefficients in the finite field $GF(q^2)$ that respect a fixed nondegenerate sesquilinear form, of determinant 1.
– PGU(n,q), projective general unitary group of $n\times n$ matrices having: coefficients in the finite field $GF(q^2)$ that respect a fixed nondegenerate sesquilinear form, modulo the centre.

– SuzukiGroup(q), Suzuki group over GF(q), $^2 B_2(2^{2k+1}) = Sz(2^{2k+1})$ .

AUTHOR:

David Joyner (2007-06): split from permgp.py (suggested by Nick Alexander)

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.

class sage.groups.perm_gps.permgroup_named.AlternatingGroup(_set)¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_symalt

class sage.groups.perm_gps.permgroup_named.CyclicPermutationGroup(n)¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

as_AbelianGroup()¶

Returns the corresponding Abelian Group instance.

EXAMPLES:: sage: C = CyclicPermutationGroup(8) sage: C.as_AbelianGroup() Multiplicative Abelian Group isomorphic to C8

is_abelian()¶

Return True if this group is abelian.

EXAMPLES:: sage: C = CyclicPermutationGroup(8) sage: C.is_abelian() True

is_commutative()¶

Return True if this group is commutative.

EXAMPLES:: sage: C = CyclicPermutationGroup(8) sage: C.is_commutative() True

class sage.groups.perm_gps.permgroup_named.DiCyclicGroup(n)¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

The dicyclic group of order $4n$ , for $n\geq 2$ .

INPUT:

n – a positive integer, two or greater

OUTPUT:

This is a nonabelian group similar in some respects to the dihedral group of the same order, but with far fewer elements of order 2 (it has just one). The permutation representation constructed here is based on the presentation

$\langle a, x\mid a^{2n}=1, x^{2}=a^{n}, x^{-1}ax=a^{-1}\rangle$

For $n=2$ this is the group of quaternions ( ${\pm 1, \pm I,\pm J, \pm K}$ ), which is the nonabelian group of order 8 that is not the dihedral group $D_4$ , the symmetries of a square. For $n=3$ this is the nonabelian group of order 12 that is not the dihedral group $D_6$ nor the alternating group $A_4$ . This group of order 12 is also the semi-direct product of of $C_2$ by $C_4$ , $C_3\rtimes C_4$ . [CONRAD2009]

When the order of the group is a power of 2 it is known as a “generalized quaternion group.”

IMPLEMENTATION:

The presentation above means every element can be written as $a^{i}x^{j}$ with $0\leq j<2n$ , $j=0,1$ . We code $a^i$ as the symbol $i+1$ and code $a^{i}x$ as the symbol $2n+i+1$ . The two generators are then represented using a left regular representation.

EXAMPLES:

A dicyclic group of order 384, with a large power of 2 as a divisor:

sage: n = 3*2^5
sage: G = DiCyclicGroup(n)
sage: G.order()
384
sage: a = G.gen(0)
sage: x = G.gen(1)
sage: a^(2*n)
()
sage: a^n==x^2
True
sage: x^-1*a*x==a^-1
True

A large generalized quaternion group (order is a power of 2):

sage: n = 2^10
sage: G=DiCyclicGroup(n)
sage: G.order()
4096
sage: a = G.gen(0)
sage: x = G.gen(1)
sage: a^(2*n)
()
sage: a^n==x^2
True
sage: x^-1*a*x==a^-1
True

Just like the dihedral group, the dicyclic group has an element whose order is half the order of the group. Unlike the dihedral group, the dicyclic group has only one element of order 2. Like the dihedral groups of even order, the center of the dicyclic group is a subgroup of order 2 (thus has the unique element of order 2 as its non-identity element).

sage: G=DiCyclicGroup(3*5*4)
sage: G.order()
240
sage: two = [g for g in G if g.order()==2]; two
[(1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)]
sage: G.center().order()
2

For small orders, we check this is really a group we do not have in Sage otherwise.

sage: G = DiCyclicGroup(2)
sage: H = DihedralGroup(4)
sage: G.is_isomorphic(H)
False
sage: G = DiCyclicGroup(3)
sage: H = DihedralGroup(6)
sage: K = AlternatingGroup(6)
sage: G.is_isomorphic(H) or G.is_isomorphic(K)
False

REFERENCES:

[CONRAD2009] Groups of order 12. Keith Conrad, accessed 21 October 2009.

AUTHOR:

Rob Beezer (2009-10-18)

is_abelian()¶

Return True if this group is abelian.

EXAMPLES:

sage: D = DiCyclicGroup(12)
sage: D.is_abelian()
False

is_commutative()¶

Return True if this group is commutative.

EXAMPLES:

sage: D = DiCyclicGroup(12)
sage: D.is_commutative()
False

class sage.groups.perm_gps.permgroup_named.DihedralGroup(n)¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

class sage.groups.perm_gps.permgroup_named.KleinFourGroup¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

class sage.groups.perm_gps.permgroup_named.MathieuGroup(n)¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

class sage.groups.perm_gps.permgroup_named.PGL(n, q, name='a')¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_plg

class sage.groups.perm_gps.permgroup_named.PGU(n, q, name='a')¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_pug

class sage.groups.perm_gps.permgroup_named.PSL(n, q, name='a')¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_plg

ramification_module_decomposition_hurwitz_curve()¶

Helps compute the decomposition of the ramification module for the Hurwitz curves X (over CC say) with automorphism group G = PSL(2,q), q a “Hurwitz prime” (ie, p is $\pm 1 \pmod 7$ ). Using this computation and Borne’s formula helps determine the G-module structure of the RR spaces of equivariant divisors can be determined explicitly.

The output is a list of integer multiplicities: [m1,...,mn], where n is the number of conj classes of G=PSL(2,p) and mi is the multiplicity of pi_i in the ramification module of a Hurwitz curve with automorphism group G. Here IrrRepns(G) = [pi_1,...,pi_n] (in the order listed in the output of self.character_table()).

REFERENCE: David Joyner, Amy Ksir, Roger Vogeler,: “Group representations on Riemann-Roch spaces of some Hurwitz curves,” preprint, 2006.
EXAMPLES:: sage: G = PSL(2,13) sage: G.ramification_module_decomposition_hurwitz_curve() #random [0, 7, 7, 12, 12, 12, 13, 15, 14]

This means, for example, that the trivial representation does not occur in the ramification module of a Hurwitz curve with automorphism group PSL(2,13), since the trivial representation is listed first and that entry has multiplicity 0. The “randomness” is due to the fact that GAP randomly orders the conjugacy classes of the same order in the list of all conjugacy classes. Similarly, there is some randomness to the ordering of the characters.

If you try to use this function on a group PSL(2,q) where q is not a (smallish) “Hurwitz prime”, an error message will be printed.

ramification_module_decomposition_modular_curve()¶

Helps compute the decomposition of the ramification module for the modular curve X(p) (over CC say) with automorphism group G = PSL(2,q), q a prime > 5. Using this computation and Borne’s formula helps determine the G-module structure of the RR spaces of equivariant divisors can be determined explicitly.

The output is a list of integer multiplicities: [m1,...,mn], where n is the number of conj classes of G=PSL(2,p) and mi is the multiplicity of pi_i in the ramification module of a modular curve with automorphism group G. Here IrrRepns(G) = [pi_1,...,pi_n] (in the order listed in the output of self.character_table()).

REFERENCE: D. Joyner and A. Ksir, ‘Modular representations: on some Riemann-Roch spaces of modular curves $X(N)$’, Computational Aspects of Algebraic Curves, (Editor: T. Shaska) Lecture Notes in Computing, WorldScientific, 2005.)
EXAMPLES:: sage: G = PSL(2,7) sage: G.ramification_module_decomposition_modular_curve() ## random [0, 4, 3, 6, 7, 8]

This means, for example, that the trivial representation does not occur in the ramification module of X(7), since the trivial representation is listed first and that entry has multiplicity 0. The “randomness” is due to the fact that GAP randomly orders the conjugacy classes of the same order in the list of all conjugacy classes. Similarly, there is some randomness to the ordering of the characters.

sage.groups.perm_gps.permgroup_named.PSP¶: alias of PSp

class sage.groups.perm_gps.permgroup_named.PSU(n, q, name='a')¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_pug

class sage.groups.perm_gps.permgroup_named.PSp(n, q, name='a')¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_plg

class sage.groups.perm_gps.permgroup_named.PermutationGroup_plg(gens=None, gap_group=None, canonicalize=True)¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

base_ring()¶

EXAMPLES:

sage: G = PGL(2,3) sage: G.base_ring() Finite Field of size 3

sage: G = PSL(2,3) sage: G.base_ring() Finite Field of size 3

matrix_degree()¶

EXAMPLES:: sage: G = PSL(2,3) sage: G.matrix_degree() 2

class sage.groups.perm_gps.permgroup_named.PermutationGroup_pug(gens=None, gap_group=None, canonicalize=True)¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_plg

field_of_definition()¶

EXAMPLES:: sage: PSU(2,3).field_of_definition() Finite Field in a of size 3^2

class sage.groups.perm_gps.permgroup_named.PermutationGroup_symalt(gens=None, gap_group=None, canonicalize=True)¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

This is a class used to factor out some of the commonality in the SymmetricGroup and AlternatingGroup classes.

set()¶

Returns the list of positive integers on which this group acts.

EXAMPLES:: sage: SymmetricGroup(3).set() (1, 2, 3) sage: SymmetricGroup([2,3,4]).set() (2, 3, 4) sage: AlternatingGroup(3).set() (1, 2, 3) sage: AlternatingGroup([2,3,4]).set() (2, 3, 4)

class sage.groups.perm_gps.permgroup_named.PermutationGroup_unique(gens=None, gap_group=None, canonicalize=True)¶: Bases: sage.structure.unique_representation.UniqueRepresentation, sage.groups.perm_gps.permgroup.PermutationGroup_generic

class sage.groups.perm_gps.permgroup_named.QuaternionGroup¶

Bases: sage.groups.perm_gps.permgroup_named.DiCyclicGroup

The quaternion group of order 8.

OUTPUT:: The quaternion group of order 8, as a permutation group. See the DiCyclicGroup class for a generalization of this construction.

EXAMPLES:

The quaternion group is one of two non-abelian groups of order 8, the other being the dihedral group $D_4$ . One way to describe this group is with three generators, $I, J, K$ , so the whole group is then given as the set $\{\pm 1, \pm I, \pm J, \pm K\}$ with relations such as $I^2=J^2=K^2=-1$ , $IJ=K$ and $JI=-K$ .

The examples below illustrate how to use this group in a similar manner, by testing some of these relations. The representation used here is the left-regular representation.

sage: Q = QuaternionGroup()
sage: I = Q.gen(0)
sage: J = Q.gen(1)
sage: K = I*J
sage: [I,J,K]
[(1,2,3,4)(5,6,7,8), (1,5,3,7)(2,8,4,6), (1,8,3,6)(2,7,4,5)]
sage: neg_one = I^2; neg_one
(1,3)(2,4)(5,7)(6,8)
sage: J^2 == neg_one and K^2 == neg_one
True
sage: J*I == neg_one*K
True
sage: Q.center().order() == 2
True
sage: neg_one in Q.center()
True

AUTHOR:: – Rob Beezer (2009-10-09)

class sage.groups.perm_gps.permgroup_named.SuzukiGroup(q, name='a')¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

base_ring()¶

EXAMPLES:: sage: G = SuzukiGroup(32, name=’alpha’) sage: G.base_ring() Finite Field in alpha of size 2^5

class sage.groups.perm_gps.permgroup_named.SymmetricGroup(_set)¶

Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_symalt

index_set(*args, **kwds)¶

Indexing sets of descent of the symmetric group.

EXAMPLES:

sage: S8 = SymmetricGroup(8)
sage: S8.index_set()
(1, 2, 3, 4, 5, 6, 7)

sage: S = SymmetricGroup([3,1,4,5])
sage: S.index_set()
(3, 1, 4)

simple_reflection(i)¶

For $i$ in the index set of self, this returns the elementary transposition $s_i=(i,i+1)$ .

EXAMPLES:

sage: A = SymmetricGroup(5)
sage: A.simple_reflection(3)
(3,4)

sage: A = SymmetricGroup([2,3,7])
sage: A.simple_reflections()
Finite family {2: (2,3), 3: (3,7)}

class sage.groups.perm_gps.permgroup_named.TransitiveGroup(d, n)¶: Bases: sage.groups.perm_gps.permgroup_named.PermutationGroup_unique

sage.groups.perm_gps.permgroup_named.TransitiveGroups(d=None)¶

INPUT:

d – an integer (optional)

Returns the set of all transitive groups of a given degree d. If d is not specified, it returns the set of all transitive groups.

Warning: TransitiveGroups requires the optional GAP database package. Please install it with sage -i database_gap.

EXAMPLES:

sage: TransitiveGroups(3)
Transitive Groups of degree 3
sage: TransitiveGroups(7)
Transitive Groups of degree 7
sage: TransitiveGroups(8)
Transitive Groups of degree 8

sage: TransitiveGroups()
Transitive Groups

Warning

in practice, the database currently only contains transitive groups up to degree 30:

sage: TransitiveGroups(31).cardinality() # requires optional database_gap
...
NotImplementedError: Only the transitive groups of order less than 30 are available in GAP's database

class sage.groups.perm_gps.permgroup_named.TransitiveGroupsAll¶

Bases: sage.sets.disjoint_union_enumerated_sets.DisjointUnionEnumeratedSets

The infinite set of all transitive groups.

EXAMPLES:

sage: L = TransitiveGroups(); L
Transitive Groups
sage: L.category()
Category of infinite enumerated sets
sage: L.cardinality()
+Infinity

sage: p = L.__iter__()            # requires optional database_gap
sage: (p.next(), p.next(), p.next(), p.next(), p.next(), p.next(), p.next(), p.next()) # requires optional database_gap
(Transitive group number 1 of degree 0, Transitive group number 1 of degree 1, Transitive group number 1 of degree 2, Transitive group number 1 of degree 3, Transitive group number 2 of degree 3, Transitive group number 1 of degree 4, Transitive group number 2 of degree 4, Transitive group number 3 of degree 4)

TESTS:

sage: TestSuite(TransitiveGroups()).run() # requires optional database_gap # long time

class sage.groups.perm_gps.permgroup_named.TransitiveGroupsOfDegree(n)¶

Bases: sage.structure.unique_representation.UniqueRepresentation, sage.structure.parent.Parent

The set of all transitive groups of a given (small) degree.

EXAMPLES:

sage: S = TransitiveGroups(4); S       # requires optional database_gap
Transitive Groups of degree 4
sage: list(S)                          # requires optional database_gap
[Transitive group number 1 of degree 4, Transitive group number 2 of degree 4, Transitive group number 3 of degree 4, Transitive group number 4 of degree 4, Transitive group number 5 of degree 4]

sage: TransitiveGroups(5).an_element() # requires optional database_gap
Transitive group number 1 of degree 5

We write the cardinality of all transitive groups of degree 5:

sage: for G in TransitiveGroups(5):    # requires optional database_gap
...       print G.cardinality()
5
10
20
60
120

TESTS:

sage: TestSuite(TransitiveGroups(3)).run() # requires optional database_gap

cardinality(*args, **kwds)¶

Returns the cardinality of self, that is the number of transitive groups of a given degree.

EXAMPLES:

sage: TransitiveGroups(0).cardinality()                      # requires optional database_gap
1
sage: TransitiveGroups(2).cardinality()                      # requires optional database_gap
1
sage: TransitiveGroups(7).cardinality()                      # requires optional database_gap
7
sage: TransitiveGroups(12).cardinality()                     # requires optional database_gap
301
sage: [TransitiveGroups(i).cardinality() for i in range(11)] # requires optional database_gap
[1, 1, 1, 2, 5, 5, 16, 7, 50, 34, 45]

Warning

The database_gap contains all transitive groups up to degree 30:

sage: TransitiveGroups(31).cardinality()                     # requires optional database_gap
...
NotImplementedError: Only the transitive groups of order less than 30 are available in GAP's database

TESTS:

sage: type(TransitiveGroups(12).cardinality())               # requires optional database_gap
<type 'sage.rings.integer.Integer'>
sage: type(TransitiveGroups(0).cardinality())
<type 'sage.rings.integer.Integer'>

“Named” Permutation groups (such as the symmetric group, S_n)¶

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