# Statement(s) > [!NOTE] Statement 1 (Homomorphism of Groups Preserves Integer Powers) > Let $G,H$ be [[Groups|groups]] and $\phi:G\to H$ a [[Homomorphism|homomorphism]]. Then for all $g\in G,$ $m\in \mathbb{Z},$ $\phi(g^m)=\phi(g)^m$where $g^m$ denotes the $m$th [[Integer Power of Group Element|power]] of $g$. **Remark**: Using additive notation, we may write $\phi(mg)=m\phi(g).$ # Proof(s) **Proof of statement 1:** Follows by induction on $m\geq 0.$ By [[Inverse of Power of Group Element]], the statement is true for all $m\in \mathbb{Z}.$ $\blacksquare$ # Application(s) **Consequences**: **Examples**: # Bibliography