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