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 .
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 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  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  and , (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.