The purpose of this paper is to study the consequences of an endomorphism near-ring of a finite group being a local near-ring and the existence of such near-rings. As we shall see in Section 2, an endomorphism near-ring of a finite group being local gives us some information about both the structure of the group (Theorem 2.2) and the automorphisms of the group lying in the near-ring (Theorem 2.3). Existence of local endomorphism near-rings of finite groups is considered in Section 3 where we obtain as our main result that any p-group of automorphisms of a p-group containing the inner automorphisms always generates a local endomorphism near-ring. In particular, we get as a corollary that the endomorphism near-ring of a finite group G generated by the inner automorphisms of G is local if and only if G is a p-group. The third section concludes with a discussion of endomorphism near-rings of dihedral 2-groups and generalized quaternion groups. PreliminariesWe shall follow the conventions of [7] with regard to notation and terminology, while our basic reference on local near-rings is [6] suitably modified to the aforementioned specifications. In particular, this means that a near-ring R is local if the set L = {reR\r does not have a right inverse} is a right i?-subgroup of R.

The study of near-rings is motivated by consideration of the system generated by the endomorphisms of a (not necessarily commutative) group. Such endomorphism near-rings also furnish the motivation for the concept of a distributively generated (d.g.) near-ring. Although d.g. near-rings have been extensively studied, little is known about the structure of endomorphism near-rings. In this paper results are presented which enable one to give the elements of the endomorphism near-ring of a given group. Also, some results relating to the right ideal structure of an endomorphism near-ring are presented. These concepts are applied to present a detailed picture of the properties of the endomorphism near-ring of (5 3 , +). PreliminariesA near-ring is a triple (R, +, .) such that (/?, +) is a group, (R, .) is a semigroup, and . is left distributive over + ; i.e. w(x+z) = wx+wz for each w,x,zeR.A near-ring R is d.g. if there exists ScR such that (S, .) is a subsemigroup of (R, .), each element of S is right distributive, and 5 is an additive generating set for (R, +). The near-ring generated additively by all the endomorphisms of a (not necessarily commutative) group (G, +) is d.g., S being the set of endomorphisms. Such a near-ring will be called an endomorphism near-ring and will be denoted by E(G).A subset K of a near ring R is an ideal if (K, +) is a normal subgroup of (R, +), rk e K, and (r l +k)r 2 -r 1 r 2 eK for each r, r u r 2 e R and k e K. A. Frohlich (5) has noted that for d.g. near-rings the third condition is equivalent to kr e K. A subset K is a right (left) ideal if K satisfies the first and third (second) conditions. Frohlich ((6), 2.4) has shown that the near-ring generated by all the inner automorphisms of a finite simple, non-commutative, group (G, +) is E(G). In fact, this near-ring generated by the inner automorphisms consists of all the mappings of G into G which leave 0 fixed. A. J. Chandy (4) has given a necessary and sufficient condition that the near-ring generated by the inner automorphisms of a group be a ring. However, the more general endomorphism near-ring has not been studied.If a is an endomorphism of (G, +) and g eG, the image of g under a is denoted by gcc. Addition of functions on G is done pointwise and multiplication of such functions is composition.

Abstract. An /-E group is a group G in which the endomorphism near-ring generated by the inner automorphisms of G equals the endomorphism nearring generated by the endomorphisms of G. In this paper we obtain a result characterizing when a semidirect product of /-E groups of relatively prime orders is an /-E group. We then use this result to show that a semidirect product of cyclic groups of relatively prime orders is an /-E group.

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