Hi. I haven't been here in a while, but I've just run across a question that I can neither find an easy counterexample for nor prove easily. Here it is: Let G be a group, and [itex]\alpha : G \rightarrow G[/itex] and [itex]\beta : G \rightarrow G[/itex] be endomorphisms. Assume that [itex]\alpha \circ \beta[/itex] is an automorphism of [itex]G[/itex]. Prove that [itex]\alpha[/itex] is injective and [itex]\beta[/itex] is surjective. If G is finite the result is easy. I do not know how to prove it for infinite groups, nor have I been able to find a simple counterexample (it's easy to construct automorphisms that are compositions and for which the composing functions don't fill the injective/surjective criteria above, but I can't find a composition of homomorphisms that do it). Thanks in advance for your help.