# Isomorphism Preserves Groups

## Theorem

Let $\left({S, \circ}\right)$ and $\left({T, *}\right)$ be algebraic structures.

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 1

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

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

From Isomorphism 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.

$\blacksquare$

## Proof 2

An isomorphism is an epimorphism.

The result follows as a direct corollary of Epimorphism Preserves Groups.

$\blacksquare$