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## Main Question or Discussion Point

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

Thanks in advance for your help.

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.