Epimorphism Preserves Groups

Theorem
Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be an epimorphism.

If $$\left({S, \circ}\right)$$ is a group, then so is $$\left({T, *}\right)$$.

Corollary
Isomorphism preserves groups.

Let $$\phi: \left({S, \circ}\right) \to \left({T, *}\right)$$ be an isomorphism.

If $$\left({S, \circ}\right)$$ is a group, then so is $$\left({T, *}\right)$$.

Proof

 * From Epimorphism Preserves Semigroups, if $$\left({S, \circ}\right)$$ is a semigroup then so is $$\left({T, *}\right)$$.


 * From Epimorphism Preserves Identity, if $$\left({S, \circ}\right)$$ has an identity $$e_S$$, then $$\phi \left({e_S}\right)$$ is the identity for $$*$$.


 * From Epimorphism Preserves Inverses, if $$x^{-1}$$ is an inverse of $$x$$ for $$\circ$$, then $$\phi \left({x^{-1}}\right)$$ is an inverse of $$\phi \left({x}\right)$$ for $$*$$.

The result follows from the definition of group.

Proof of Corollary
An isomorphism is an epimorphism.

Hence the result.