We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic , then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be torsion fre...

A set of quasi-uniform random variables X1, . . . , Xn may be generated from a finite group G and n of its subgroups, with the corresponding entropic vector depending on the subgroup structure of G. It is known that the set of entropic vectors obtained by considering arbitrary finite groups is much richer than the one provided just by abelian groups. In this paper, we start to investigate in mo...

We establish a correspondence between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying field has positive characteristic, then an abelian regular subgroup has finite exponent if the vector space is finite-dimensional, while it can be to...

Let G be a finite abelian group, written additively, and H subgroup of G. The sum graph $$\varGamma _{G,H}$$ is the with vertex set G, in which two distinct vertices x y are joined if $$x+y\in H{\setminus }\{0\}$$ . These graphs form fairly large class Cayley graphs. Among cases have been considered previously prime graphs, case where $$H=pG$$ for some number p. In this paper we present their s...

A finite group is called simple when its only normal subgroups are the trivial subgroup and the whole group. For instance, a finite group of prime size is simple, since it in fact has no non-trivial proper subgroups at all (normal or not). A finite abelian group G not of prime size, is not simple: let p be a prime factor of #G, so G contains a subgroup of order p, which is a normal since G is a...

We conclude thhat T is a subgroup. Now we exhibit an example where T fails to be a subgroup when G is non-abelian. Let G = 〈x, y|x = y = i 〉. Observe that the product xy has infinite order since the product (xy)(xy) is no longer reducible because we lack the commutativity of x with y. Hence, x, y ∈ T but (xy) / ∈ T . Exercise (7). Fix some n ∈ Z with n > 1. Find the torsion subgroup of Z×(Z/nZ)...