Laws of exponents in a group

Statement

Let $G$ be a group with identity element $e$, and let $a \in G$. For an integer $n$, define $a^n$ as follows: $a^0 = e$, $a^{n+1} = a^n a$ for $n \geq 0$, and $a^{-n} = (a^{-1})^n$ for $n > 0$. Then $a^{m+n} = a^m a^n$ and $(a^m)^n = a^{mn}$ for all integers $m$ and $n$.

Remarks

The law $(ab)^n = a^n b^n$ is true for all $n$ when $ab = ba$, but it can fail when $a$ and $b$ do not commute. In additive notation the laws are $(m + n)a = ma + na$ and $n(ma) = (nm)a$. The first law shows that $n \mapsto a^n$ is a group homomorphism from $(\mathbb{Z}, +)$ to $G$.

Explanations

L1 human

The powers of an element of a group follow the rules that you know for the powers of a nonzero number. A negative power means a power of the inverse, so $a^{-2} = a^{-1} a^{-1}$. For example, $a^3 a^{-1} = a a a a^{-1} = a^2$.