# Definition:Cycle Type

Jump to navigation
Jump to search

## Definition

Let $S_n$ denote the symmetric group on $n$ letters.

Let $\rho \in S_n$.

From Existence and Uniqueness of Cycle Decomposition, every $\rho$ may be uniquely expressed as a product of disjoint cycles:

- $\rho = \tau_1, \tau_2, \ldots, \tau_r$

up to the order of factors.

Let $\tau_1, \tau_2, \ldots, \tau_r$ be arranged in increasing order of cycle length.

Let the length of the cycle $\tau_i$ be $k_i$.

The resulting ordered tuple of cycle lengths $\tuple {k_1, k_2, \ldots, k_r}$ is called the **cycle type** of $\rho$.

Thus $\sigma$ and $\rho$ have the same **cycle type** if they have the same number of cycles of equal length.

## Also known as

Some sources refer to the **cycle type** of a permutation as its **form**.

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: The Symmetric Groups: $\S 80 \alpha$ - 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $9$: Permutations: Proposition $9.20$: Remark