They also determined that G(H, [k.sub.1]) [??] G(H, [k.sub.2]) for an

abelian group H by using graph theory and group theory.

Since each finite

Abelian group and dihedral groups are determined by their endomorphism monoids in the class of all groups (Lemmas 2.11 and 2.13), we have

Let G be an

abelian group and 0 [not equal to] [alpha] [member of] G be a fixed element.

G is solvable if it has a subnormal series so that [G.sub.i+1]/[G.sub.i] is an

abelian group. Generalizing this definition to Hopf algebras requires an appropriate translation for subnormal series in addition to abelian quotients.

Let H be a finite

abelian group and f [member of] End(H).

Then, a joint determinant may be thought of as a map from Comm((k) into an

abelian group G.

Let G be a compact

Abelian group. Then, on (G, b(G)), the probability Haar measure [mu] can be defined.

Denote by [[chi].sub.W] : [??] [right arrow] [S.sup.1] the characters of the irreducible representations W of the

abelian group [??], and write x = [[summation].sub.W[member of]S([??])] [y.sup.W.sub.n] in terms of the decomposition on isotypical components [mathematical expression not reproducible] relative to [[??]'.sub.n] with [mathematical expression not reproducible].

For a commutative ring R with identity 1 and a finite

abelian group G, written additively, let R[G] denote the group ring of G over R.

Moreover, the notion of bond lattice comes from the study of Galois connections, and a natural action of set partitions that is analogous to the action of integers on any

abelian group. Naturally, in order to say that set partitions act on abelian Hopf monoids, we need to define the notion of ring in species.

As we will see, he can follow the same strategy in any

abelian group.

A subgroup H of an

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

The Group Calculator allows the user to select a group from one of several families: cyclic group of order n, dihedral group [D.sub.n] of order 2n, the group [Z.sup.*.sub.n] of units modulo n, the

abelian group [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and the affine group Aff([Z.sub.n]) = {ax + b | a [member of] [Z.sup.*.sub.n] and b [member of] [Z.sub.n]} under composition.

Sooryanarayana, Hamiltonian Distance Generating sets of an

Abelian Group, Proceedings, National Seminar on Recent developments in applications of Mathematics held at Sri Padmavathi Mahila University, Tirupati, Andhra Pradesh, India, during 21-22, March 2005.

In this note, we will show that the set of functions [[xi].sub.[alpha]] ([alpha] [member of] C) forms an

Abelian group with the Dirichlet series multiplication followed by a number of applications.