Martin Newell

Normal and power endomorphisms of a group

A normal endomorphism of a group G is an endomorphism which commutes with every inner automorphism of G. Thus the image of a normal subgroup of G under a normal endomorphism is itself a normal subgroup of G. Following [1] we call an endomorphism of G which maps every subgroup of G into itself a power endomorphism of G, Under a power endomorphism of G every element is mapped to some power. In this note we show that every power endomorphism 0 of G is necessarily normal (Theorem 2). In particular this implies that G ~ -o) is contained in the centre Z(G) of G. If in addition 0 is an epimorphism of G, then G(Z-0) is contained in Z(G) and thus one obtains an improvement of [1] Theorem 2.2.1. Our notation will be that of [1]. In addition for endomorphisms 0 and 4) of a group G we shall write 0-~b for the mapping: g_~gOg-4, where g~G. Ifg, h~G, then

ON A CLASS OF NORMAL ENDOMORPHISMS OF GROUPS

It is proved that if θ is an endomorphism of a group G such that 〈x, xθ〉 is cyclic for all elements x of G, then θ is a normal endomorphism of G, i.e. it commutes with all inner automorphisms of G.

Infinite groups 1994 : proceedings of the international conference held in Ravello, Italy, May 23-27, 1994

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