Group theory and generalizations Proposition

Laws of exponents in a group

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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 $G$ .

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Laws of exponents in a group statement

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Tag	`00DRK90Z6K`
URI	`m2://00DRK90Z6K`
Aliases	`power-laws-in-a-group`
Kind	statement, proposition
Writing	W0 stub
MSC	20A05
Status	This copy of the site has no build data. Run `m2 status 00DRK90Z6K`.
Uses	Group

Laws of exponents in a group

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