| Popis: |
Let (G,+) be a group, not necessarily abelian, and let K be a nonzero subgroup of G. Let H = H (G, K) be the additive group generated by H om(G, K). Then H is a d.g. near-ring. This near-ring can be a ring even when \( \bar{K} \), the additive subgroup generated by ∪Gα, α ∈ H is nonabelian. Conditions on K or \( \bar{K} \) are given for this to occur, with examples to illustrate the theory developed. Similarly conditions are developed for H to be distributive. The function (α)Θ = α∣K, α ∈H, plays a key role in this study. The mapping Θ is a near-ring homomorphism from H(G, K) into the d.g. near-ring e(K), the near-ring generated by End K. Properties of H(G, K), where K or \( \bar{K} \) is abelian, are investigated. There is a rich interplay between the near-rings (rings) H(G, K) and e(K) and the groups G, K, and \( \bar{K} \). Exemplary of this is: let K be abelian, then Θ is surjective if and only if G is a semidirect sum of K and a normal subgroup. |
