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Let G be a finite group and H be a Hartley set of G. In this paper, we prove the existence and conjugacy of H -injectors of G and describe the characterization of injectors via radicals. As applications, some known results are directly followed. [ABSTRACT FROM AUTHOR]

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd(p, k) = 1. [ABSTRACT FROM AUTHOR]