Examples of such groups are: finite Abelian groups
(, Theorem 4.2), generalized quaternion groups (, Corollary 1), torsion-free divisible Abelian groups (, Theorem 1), etc.
Let H be a finite abelian group. By the structure of finite abelian group, we have
Let [H.sub.1], [H.sub.2] be two finite abelian groups and [f.sub.i] [member of] End([H.sub.i]).
because each finite Abelian group
is determined by its endomorphism semigroup in the class of all groups (, Theorem 4.2).
Since (Q, +) and (Q', +') are finite Abelian groups
, their isomorphism follows from End(Q, +) [congruent to] End( Q', +') (see Theorem 4.2 in ).
A quasi-arithmetic matroid endows a matroid with a multiplicity function, whose values (in the representable case) are the cardinalities of certain finite abelian groups, namely, the torsion parts of the quotients of an ambient lattice [Z.sup.n] by the sublattices spanned by subsets of vectors.
Some significant information about an integer vector configuration is not retained in the multiplicity function, as many finite abelian groups can have the same cardinality.
Note that this includes the class of all finite abelian groups
, because, if G is abelian of order g and H is a subgroup of order h, then [Gamma](G, H) is disconnected with exactly h components each component being complete with g/h vertices.
Let a(n) denote the number of non-isomorphic finite abelian groups
with n elements.
Papers cover such subjects as outer automorphism groups of certain orientable Seifert three-manifold groups, a proposed public key cryptosystem using the modular group, normal subgroups of themodular group and other Hecke groups, unions of varieties and quasi-varieties, context-free irreducible word problems in groups, informative words and discreteness, using group theory for knowledge representation and discovery, torsion in maximal arithmetic Fuchsian groups, density of test elements in finite Abelian groups
and the Rosenberg "monster."