Every finite

Abelian group is determined by its endomorphism monoid in the class of all groups.

Then we exploit the fact that for a nondegenerate pairing on an

abelian group holds [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and that any r [member of] S acts on v by the 1-dimensional character [[chi].sub.j](r) = (r,[s.sub.j]):

Let H be a finite

abelian group written additively and End(H) be the endomorphism ring of H.

Singh, "Some decomposition theorems on

abelian groups and their generalisations.

Corollary 1.2 If G is an

abelian group of order n and n is divisible by m then there exists a subgroup of G with order m.

In fact the two objects are not truly equivalent, in that the information contained in matroids over Z is richer, because there are many finite

abelian groups with the same cardinality.

A subgroup H of an

abelian group G is pure in G if nH = H (1 nG, where n is any non-zero integer.

Rankin, On the number of

abelian groups of a given order, Quart J.

Throughout this brief article, let G be an

abelian group with p-component of torsion [G.sub.p] and R a commutative ring with 1 (called also unital) of prime characteristic p with nil-radical N(R).

These very different groups R, [T.sup.1], Z are all abelian and locally compact ([T.sup.1] is compact), and so are united by working with locally compact

abelian groups.

This note shows how to obtain an

abelian group with an addition like operation (Joyner 2002:70-72) beginning with a subtraction binary operation.