Group theory and generalizations Proposition

g to the n equals 1 iff the order divides n

Builds on Order of an element Laws of exponents in a group

Statement

Let $G$ be a group with identity element $e$ , and let $g \in G$ be an element of finite order $m$ . Let $n$ be an integer. Then $g^n = e$ if and only if $m$ divides $n$ .

Remarks

The proof divides $n$ by $m$ with remainder: $n = qm + r$ with $0 \leq r < m$ , so that $g^n = (g^m)^q g^r = g^r$ . By the minimality of $m$ , the power $g^r$ is $e$ only when $r = 0$ . A consequence is that $g^a = g^b$ if and only if $a \equiv b \pmod m$ .

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Group concept
Laws of exponents in a group statement
Order of an element definition
g to the n equals 1 iff the order divides n statement

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21 pages build on this one.

Tag	`0F5QC101Q7`
URI	`m2://0F5QC101Q7`
Aliases	`power-is-identity-iff-order-divides`
Kind	statement, proposition
Writing	W0 stub
MSC	20A05
Status	This copy of the site has no build data. Run `m2 status 0F5QC101Q7`.
Uses	Order of an element, Laws of exponents in a group

g to the n equals 1 iff the order divides n

Statement

Remarks

Reading path

Where this leads

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