Group homomorphism

Group epimorphism

Let be a homomorphism. The following two statements are equivalent¹: group

• is surjective
• is epic in Category of groups

Proof

Since all surjective functions are epic, it follows that surjective homomorphisms are too. The converse requires more care. Let be an epimorphism and . It is sufficient to show . Consider the set of left-cosets of in , and let be a unique symbol, , and denote the permutation group of the set . We define a homomorphism so that for any

Let be the permutation which swaps and and leaves everything else invariant, and let be its induced inner automorphism. Then is also a homomorphism. Now, for any it follows leaves both and fixed (the latter is by construction), and hence commutes with and thus . Hence , and since is epic . But thence follows that commutes with for all , wherefore must leave fixed, and thus and thus for all .

Corollaries

• The above argument holds for Category of finite groups, since is finite if and are finite.

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Footnotes

1. 1970. A Group Epimorphism is Surjective ↩