Groups in which each element commutes with its endomorphic images

Abstract

Let G be a finite group in which each element commutes with its endomorphic images. We will give a counterexample to the conjecture that G is abelian. Let G be a finite group in which each element commutes with its endomorphic images. We will give a counterexample to the conjecture that G is abelian. For g and h in G, g and g-1(h-1lgh) commute. Thus G satisfies the identical relation (x, y, x) = 1. Therefore G satisfies the identical relations (x, y, z, w)=1 and (x, y, z)3=I (see [1, p. 322]). Hence G is nilpotent of class <3 and if G is a p-group for a prime ps3, G is nilpotent of class <2. We will exhibit for any prime p a nilpotent class 2 p-group in which each element commutes with all of its endomorphic images. Let p2 G = (a,, a2, a3, a4:a = 1, (ai, aj, ak) = 1, (1 < i,j, k < 4) p p ~~~~~~~~p and (1) (a,, a2) = al, (2) (a,, a3) = a3, (3) (a1, a4) = a4, (4) (a2, a3) = a2, (5) (a2, a4) = 1, (6) (a3, a4) = a3). Let E(G) and A (G) denote the endomorphisms and automorphisms of G respectively. Let EZ(G) = {GCE(G):0(G) <Z(G) } and AZ(G) {OCA(G):O(g)g-1E'Z(G) for all gEEG } =central automorphisms of G. We will prove THEOREM. E(G) =EZ(G)kUAZ(G). COROLLARY. Each element in G commutes with its endomorphic images. LEMMA 1. G| =p PROOF. Clearly from the defining relations I GI <p8. We will construct a group of order p8 which satisfies the defining relations of G. Let Received by the editors February 11, 1970 and, in revised form, April 24, 1970. AMS 1969 subject classifications. Primary 2025, 2040.