Nearrings of functions without identity determined by a single subgroup

Let (G, +) be a finite group, written additively with identity ۰, but not necessarily abelian, and let H be a nonzero, proper subgroup of G. Then the set M = \{f : G \to G\ |\ f(G) \subseteq H \ \hbox{and}\ f(۰) = ۰ \} is a right, zero-symmetric nearring under pointwise addition and function composition. We find necessary and sufficient conditions for M to be a ring and determine all ideals of M, the center of M, and the distributive elements of M.

Keywords:

Abelian , distributive , center , ideal , zero-symmetric