# Definition:Cycle Type

## 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:

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

up to the order of factors.

Let $\tau_1, \tau_2, \ldots, \tau_l$ 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 $\tuple {k_1, k_2, \ldots, k_l}$ 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.

## Sources

- 1996: John F. Humphreys:
*A Course in Group Theory*... (previous) ... (next): Chapter $9$: Permutations: Proposition $9.20$: Remark