A covering problem over finite rings

AbstractGiven a finite commutative ring with identity A, define c(A,n,R) as the minimum cardinality of a subset H of An which satisfies the following property: every element in An differs in at most R coordinates from a multiple of an element in H. In this work, we determine the numbers c(Zm,n,0) for all integers m≥2 and n≥1. We also prove the relation c(S×A,n,1)≤c(S,n−1,0)c(A,n,1), where S=Fq or Zq and q is a prime power. As an application, an upper bound is obtained for c(Zpm,n,1), where p is a prime

A note on semidirect products and nonabelian tensor products of groups

Let G and H be groups which act compatibly on one another. In [2] and [8] it is considered a group construction η(G,H) which is related to the nonabelian tensor product G⊗H. In this note we study embedding questions of certain semidirect products A⋊H into η(A,H), for finite abelian H-groups A. As a consequence of our results we obtain that complete Frobenius groups and affine groups over finite fields are embedded into η(A,H) for convenient groups A and H. Further, on considering finite metabelian groups G in which the derived subgroup has order coprime with its index we establish the order of the nonabelian tensor square of G