Isomorphism Preserves Groups/Proof 1

Proof
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.