---
tag: 00DRK90Z6K
kind: statement
role: proposition
name: Laws of exponents in a group
aliases: [power-laws-in-a-group]
msc: [20A05]
uses: [G5S1M9]
sources:
  - url: https://en.wikipedia.org/wiki/Exponentiation
    locator: In a group
---

## Statement

Let $G$ be a [[group|G5S1M9]] 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, $a^3 a^{-1} = a a a a^{-1} = a^2$.