General Group Properties#

IsCyclic(G): GrpAb -> BoolElt#: Returns true if the group \(G\) is cyclic, false otherwise.

IsElementaryAbelian(G): GrpAb -> BoolElt#: Returns true if the group \(G\) is elementary abelian, false otherwise.

IsFree(G): GrpAb -> BoolElt#: Returns true if \(G\) is free, false otherwise.

IsMixed(G): GrpAb -> BoolElt#: Returns true if \(G\) is a mixed group, false otherwise. An abelian group is mixed if it is neither a torsion group nor free.

IspGroup(G): GrpAb -> BoolElt#: Returns true if the finite group \(G\) is a \(p\)-group, i.e. if all elements have order a power of \(p\).

DerivedLength(G): GrpAb -> RngIntElt#: The length of the derived series of \(G\).

Properties of Subgroups#

IsMaximal(G, H): GrpAb, GrpAb -> BoolElt#: Returns true if the subgroup \(H\) of the finite group \(G\) is a maximal subgroup of \(G\), false otherwise.

Index(G, H): GrpAb, GrpAb -> RngIntElt#: The index of the subgroup \(H\) in the group \(G\), returned as an ordinary integer. If \(H\) has infinite index in \(G\), the value zero is returned.

FactoredIndex(G, H): GrpAb, GrpAb -> [<RngIntElt, RngIntElt>]#: The factored index of the subgroup \(H\) in the group \(G\), returned as a sequence of prime-exponent pairs. If \(H\) has infinite index in \(G\), the empty sequence is returned.

IsPure(G, H): GrpAb, GrpAb -> BoolElt#: Returns true if the subgroup \(H\) of the finite group \(G\) is pure, ie. if for all \(n\) we have \(nG \cap H = nH\).

IsNeat(G, H): GrpAb, GrpAb -> BoolElt#: Returns true if the subgroup \(H\) of the finite group \(G\) is neat, i.e., if for all primes \(p\) we have \(pG \cap H = pH\).

Enumeration of Subgroups#

MaximalSubgroups(G): GrpAb -> [GrpAb]#: The maximal subgroups of the finite group \(G\) returned as a sequence of subgroups.

Subgroups(G:parameters): GrpAb -> [Rec]#

Sub : [RngIntElt]                    Default: []
Quot: [RngIntElt]                    Default: []

The subgroups of the finite group \(G\) are returned as a sequence of records. The record fields are subgroup, storing the actual group; order, storing the group order; and length, storing the length of the conjugacy class, which is always 1 for abelian groups.

If the parameter Sub is set, only subgroups with invariants equal to the given sequence are found. The given sequence should contain positive integers, such that each divides the following.

If the parameter Quot is set, only subgroups such that the quotient group has invariants equal to the given sequence are found. The given sequence should contain positive integers, such that each divides the following.

NumberOfSubgroupsAbelianPGroup(A): SeqEnum -> SeqEnum#: Return the number of subgroups of each non-trivial order in the abelian \(p\)-group \(G\) where \(A = [a_1, a_2, \ldots]\) and \(G = C_{a_1} \times C_{a_2} \times \ldots\). The \(m\)-th entry in the sequence returned is the number of subgroups of order \(p^m\).

HasComplement(G, U): GrpAb, GrpAb -> BoolElt, GrpAb#: For a finite abelian group \(G\) and a subgroup \(U\) decide if there exist some other subgroup \(V\) such that \(G = U+V\) and \(U \cap V = \lbrace 0\rbrace\). In case such a \(V\) exists, it is returned as the second value.

Example: Subgroups (ex-604d09)#

We look at subgroups of an abelian group of order 12.

> G := AbelianGroup([2,6]);
> s := Subgroups(G); #s;
10
> s[7];
rec<recformat<order, length, subgroup, presentation> |
   order := 3, length := 1,
   subgroup := Abelian Group isomorphic to Z/3
Defined on 1 generator in supergroup G:
  $.1 = 2*G.2
Relations:
  3*$.1 = 0>
> [x`order:x in s];
[ 12, 6, 4, 2, 6, 6, 3, 2, 2, 1 ]

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Now we find the elementary abelian subgroup of order 4.

> s22 := Subgroups(G:Sub := [2,2]); #s22;
1
> s22;
Conjugacy classes of subgroups
------------------------------

[1]     Order 4            Length 1
        Abelian Group isomorphic to Z/2 + Z/2
        Defined on 2 generators in supergroup G:
          $.1 = G.1
          $.2 = 3*G.2
        Relations:
          2*$.1 = 0
          2*$.2 = 0

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There is more than one subgroup of index 2 in \(G\).

> q2 := Subgroups(G:Quot := [2]); #q2;
3
> q2[3]`subgroup;
Abelian Group isomorphic to Z/6
Defined on 1 generator in supergroup G:
  $.1 = G.1 + G.2
Relations:
  6*$.1 = 0

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