In this section we review some basic properties of G(n, k) and G(H, k), where H is a finite

abelian group. The structure of G(H, k) and hence [G.sub.1](n, k) is well understood in [16].

Examples of such groups are: finite

Abelian groups ([12], Theorem 4.2), generalized quaternion groups ([13], Corollary 1), torsion-free divisible

Abelian groups ([16], Theorem 1), etc.

Let G be an

abelian group and let 0 denote the identity element of G.

Let H be a finite

abelian group. By the structure of finite

abelian group, we have

The main result in [1] about the joint determinants is that there exists a one-to-one correspondence between the set of joint determinants from [Comm.sub.l](k) into an

abelian group G and the set of group homomorphisms from Milnor's K-group [K.sup.M.sub.l](k) into G.

To study self-dual abelian codes, it is sufficient to focus on [mathematical expression not reproducible], where A is an

abelian group of odd order and B is a nontrivial

abelian group of two power order.

Theorem 1.1 (Fundamental theorem of finite

abelian groups) Any finite

abelian group G can be written as a direct sum of cyclic groups in the following canonical way: G = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where every [k.sub.i] (1 [less than or equal to] i [less than or equal to] l) is a prime power.

On the Neat Essential Extensions of

Abelian Group Journal of Business Strategies, 4 (1), 1-6.

Sooryanarayana., Hamiltonian Distance Generating sets of an

Abelian Group, Far East Journal of Appl.

From [2] and [5], (M, *) forms an

Abelian group with identity element [[mu].sub.0] under the operation * : M x M [??] M, [[mu].sub.[alpha]] * [[mu].sub.[beta]] = [[mu].sub.[alpha]+[beta]], where [alpha], [beta] [member of] C.

Let G be additive

Abelian group and H is a subgroup of G.