?

Abstract Algebra

Free Version

Upgrade subject to access all content

Moderate

Cycle Types

ABSALG-TNXKV5

If the disjoint cycle decomposition of a permutation $\sigma \in S_n$ has cycle lengths $a_1 \ldots a_k$, with $a_1\geq a_2 \cdots \geq a_k$, define the cycle type of $\sigma$ to be $(a_1,\ldots, a_k)$. Note that we always have $a_1 + \cdots + a_k =n$.

For example, the cycle type of $(12)(45) \in S_5$ is $(2,2,1)$.

What are the cycle types of all powers of $\sigma=(123)(45)(67) \in S_7$?

A

$(1,1,1,1,1,1,1), (3,2,2)$

B

$(3,2,2), (3,1,1,1,1), (2,2,1,1,1)$

C

$(1,1,1,1,1,1,1), (3,2,2), (3,1,1,1,1), (2,2,1,1,1)$

D

$(1,1,1,1,1,1,1), (3,2,2), (3,1,1,1,1), (2,2,1,1,1), (3,2,1,1), (3,3,1), (2,2,2,1)$