7.46 AbelianInvariants

AbelianInvariants( G )

Let G be an abelian group. Then AbelianInvariants returns the abelian invariants of G as a list of integers. If G is not abelian then the abelian invariants of the commutator factor group of G are returned.

Let G be a finitely generated abelian group. Then there exist n nontrivial subgroups A_i of prime power order p_i^{e_i} and m infinite cyclic subgroups Z_j such that <G> = A_1 times ... times A_n times Z_1 ... times Z_m. The invariants of G are the integers p_1^{e_1}, ..., p_n^{e_n} together with m zeros.

Note that AbelianInvariants tests and sets G.abelianInvariants.

    gap> AbelianInvariants( AbelianGroup( AgWords, [2,3,4,5,6,9] ) );
    [ 2, 2, 3, 3, 4, 5, 9 ] 

The default function GroupOps.AbelianInvariants requires that G is finite.

Let G be a finite abelian group of order p_1^{e_1}...p_n^{e_n} where p_i are distinct primes. The default function constructs for every prime p_i the series <G>, <G>^{p_i}, <G>^{p_i^2}, ... and computes the abelian invariants using the indices of these groups.

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