Group theory and generalizations Proposition

Order of a power

Builds on Order of an element g to the n equals 1 iff the order divides n

Statement

Let $G$ be a group and let $g \in G$ be an element of finite order $n$ . Let $k$ be an integer. Then the order of $g^k$ is $n / \gcd(n, k)$ .

Remarks

Here $\gcd(n, 0) = n$ , so that $g^0 = e$ has order $1$ . In particular, $g^k$ has order $n$ if and only if $\gcd(n, k) = 1$ . For example, if $g$ has order $12$ , then $g^8$ has order $12 / 4 = 3$ .

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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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
Order of a power statement

Where this leads

3 pages build on this one.

Tag	`0VVPQNVM8S`
URI	`m2://0VVPQNVM8S`
Kind	statement, proposition
Writing	W0 stub
MSC	20A05
Status	This copy of the site has no build data. Run `m2 status 0VVPQNVM8S`.
Uses	Order of an element, g to the n equals 1 iff the order divides n

Order of a power

Statement

Remarks

Reading path

Where this leads

Further reading

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