An interesting orbit-induced partition of the class of all groups

Abstract

Given that the set of endomorphisms of a group is contained in the set of distributive elements of its endomorphism near-ring, which, in turn, is contained in the endomorphism near-ring, we show that the class of all groups is partitioned into four nonempty subclasses when all combinations of these inclusions, proper or non-proper, are considered. Furthermore, a characterization of each subclass is given in terms of the orbits of the underlying group. Mathematics Subject Classification: 20E99

Direct product of four classes of groups

“…In the remaining cases we prove that G is a cyclic E(G)-module. Hence G is a D-H group by proposition 5.3 (Boudreaux 2007a). Let G = Q 4n , the generalized quaternion group Q 4n of order 4n.…”

“…Take f = f 1 + f 2 , then f ∈ E(G) and f (y) = f 1 (y) f 2 (y) = yx n−1 y = x. Hence G = E(G)(y) is a D-H group by proposition 5.3 (Boudreaux 2007a). …”

“…From this it follows that the second subclasses is nonempty. In Boudreaux (2007a), the author has shown the remaining two subclass of groups are nonempty. Furthermore, he gave a characterization of each subclass in terms of the orbits of the underlying group.…”

“…Since H and K are D-H groups, they are respectively cyclic E(H )-module and E(K )-module (Corollary 5.7 of Boudreaux (2007a)). Now by the proof of the preceeding theorem G is also a cyclic E(G)-module.…”

“…For more details on the structure of H(G, K ) see papers (Birkenmeier et al 1997a,b). To prove the next result we need Lemma 6.6 and Theorem 6.5 of Boudreaux (2007a).…”