AbelianGroup(Q) : [ RngIntElt ] -> GrpAb
Let Q = [ a1, ..., ar] be a sequence of non-negative integers.
This function creates the abelian group Z1 + ... + Zr, where
Zi is the cyclic group of order |ai| if aineq0 or the infinite
cyclic group Z otherwise, i = 1, ..., r.
Given an abelian permutation, matrix or polycyclic group G, represent
it as an abelian group A. The function also returns the isomorphism
φ: G -> A as its second value.
Given a finitely presented, permutation, matrix or polycyclic group
G, return the maximal abelian quotient A of G. The function
returns the natural homomorphism φ: G -> A as its second
value.
The direct sum of abelian groups A and B.
A pc-group representation G of A.
The isomorphism φ: A -> G is also returned.
A permutation group representation of A. The particular group G
is generated by disjoint cycles whose lengths are the abelian
invariants of A. The isomorphism φ: G -> A is also
returned.
A fp-group group representation of A. The particular group G
is generated by commuting generators whose orders are the abelian
invariants of A. The isomorphism φ: G -> A is also
returned.
DerivedSubgroup(G) : GrpAb -> GrpAb
DerivedGroup(G) : GrpAb -> GrpAb
The derived subgroup of G, that is the trivial group, since G is abelian.
CommutatorSubgroup(G, H, K) : GrpAb, GrpAb, GrpAb -> GrpAb
The commutator subgroup of groups H and K in their common overgroup G.
Centraliser(G, a) : GrpAb, GrpAbElt -> GrpAb
The centraliser of a in G.
The maximal normal subgroup of G that is contained in the subgroup H of
G. Since G is abelian, this is H itself.
Center(G) : GrpAb -> GrpAb
The center of G, ie. G itself.
FittingSubgroup(G) : GrpAb -> GrpAb
The Fitting subgroup of G.
Hypercenter(G) : GrpAb -> GrpAb
The hypercentre of G.
V2.28, 13 July 2023