Group theory and generalizations Proposition

Order of a permutation is the lcm of its cycle lengths

Builds on Decomposition of a permutation into disjoint cycles Order of an element Disjoint cycles commute

Statement

Let $\sigma \in S_n$ , and let $\sigma = c_1 c_2 \cdots c_r$ be its decomposition into disjoint cycles of lengths $k_1, \dots, k_r$ . Then the order of $\sigma$ is $\operatorname{lcm}(k_1, \dots, k_r)$ .

Remarks

A $k$ -cycle has order $k$ . The disjoint cycles commute, so $\sigma^m = c_1^m \cdots c_r^m$ , and this is the identity exactly when each $c_i^m$ is the identity. For example, $(1\ 2)(3\ 4\ 5)$ has order $6$ .

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Cycle notation concept
Group concept
Laws of exponents in a group statement
Symmetric group definition
Disjoint cycles commute statement
Order of an element definition
Decomposition of a permutation into disjoint cycles statement
Order of a permutation is the lcm of its cycle lengths statement

Tag	`0DH7HMXP9F`
URI	`m2://0DH7HMXP9F`
Kind	statement, proposition
Writing	W0 stub
MSC	20B30
Status	This copy of the site has no build data. Run `m2 status 0DH7HMXP9F`.
Uses	Decomposition of a permutation into disjoint cycles, Order of an element, Disjoint cycles commute

Order of a permutation is the lcm of its cycle lengths

Statement

Remarks

Reading path

Further reading

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