In this paper two major questions concerning generalised quaternion groups and distributively generated (d.g.) near-rings are investigated. The d.g. nearrings generated, respectively, by the inner automorphisms, automorphisms, and endomorphisms of the group are described. It is also shown that these morphism near-rings are local near-rings and contain no non-trivial idempotents. Finally, it is demonstrated that exactly 16 d.g. near-rings can be defined on a given generalised quaternion group. Properties of Q nThe quaternion group of order 2", n ^ 3, will be designated by Q n and will be presented as (a, b \ a 2 "' 1 , bab~la, a 2 "' 2 b 2 ). Elements of Q n will be given in the form cfb s , 0^x g 2 " " 1 -l , O g s g 1. Unless otherwise noted, it is assumed that n>3. (8), pp. 191-192, it follows that each of Sand Tis isomorphic to Q n -i and also that the subgroups of (a) are normal in Q n .Theorem 2. Q n has 2"~l inner automorphisms, 2 2 "~3 automorphisms, and 2 2 "~3 + 4 endomorphisms.Proof. The first statement follows since Q n has a centre of order 2 (see p. 192 of (8)), the second statement is given on page 133 of (1).Since Q n modulo either S or T or (a) is of order 2 and Q n contains a unique element of order 2, these normal subgroups each serve as the kernel for only one endomorphism. The subgroups of (a), other than (a) itself, lead to quotient groups which contain more than one element of order 2 and so cannot be the kernels of endomorphisms. The last endomorphism is the trivial one which has Q n as its kernel.

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Abstract. If G is a group written additively, the inner automorphisms and the endomorphisms additively generate near-rings

Groups for which the distributively generated near-ring generated by the endomorphisms is in fact a ring are known as E-groups and are discussed in (3). R. Faudree in (1) has given the only published examples of non-abelian E-groups by presenting defining relations for a family of p–groups. However, as shown in (3), Faudree's group does not have the desired property when p = 2.