Exponent Laws for Integer Powers in a Group

In the setting of Groups: Standing Notation, let $G$ be a group with identity $e$, let $g\in G$, and let $m,n\in\mathbb{Z}$.

(i) First power.

$$g^{1}=g.$$

(ii) Powers of the identity.

$$e^{n}=e.$$

(iii) Sums of exponents.

$$g^{m+n}=g^{m}g^{n}.$$

(iv) Products of exponents.

$$(g^{m})^{n}=g^{mn}.$$

(v) Inverses.

$$(g^{n})^{-1}=g^{-n}=(g^{-1})^{n}.$$

(vi) Commuting elements. Let $h\in G$ with $gh=hg$. Then

$$(gh)^{n}=g^{n}h^{n}.$$

(vii) Homomorphisms. Let $H$ be a group and $f:G\to H$ a group homomorphism. Then

$$f(g^{n})=f(g)^{n}.$$