The number of elements of order $5$ in $S_5$

Question

I need to find the number of element of order $5$ in $S_5$ and

The number of elements in $S_{10}$ commuting with $(1 3 5 7 9)$.

for the first one I have thought that it should be $5!/5=4!$, could any one help me for the second one?

Answer 1

For the second part, you could argue as follows. The number of elements commuting with $g:=(13579)$ is precisely the order of the centralizer of $g$ in $G:=S_{10}$, by its very definition. If we denote such a centralizer by $H$, notice that $H$ is exactly the stabilizer of $g$ for the conjugacy action of $G$ on itself (see here). By the Orbit-Stabilizer Theorem, $\vert G: H\vert$ (the index of $H$ in $G$) is the cardinality of the orbit of $g$ for the conjugacy action, i.e. it is the cardinality of the conjugacy class of $g$, $g^{G}$. Therefore, by Lagrange's Theorem, $\vert H\vert =\vert G\vert /\vert g^{G}\vert=10!/\vert g^{G}\vert$. But $\vert g^{G}\vert$ is simply the subset of $G$ given by the $5-$cycles in $S_{10}$, since two elements in $S_{n}$ belong to the same conjugacy class if and only if they have the same cyclic structure. At this point, can you count how many distinct $5-$cycles are there in $S_{10}$?

Answer 2

For the first question you have 120 choices for the number to put into the cycle (5 for the first,4 for the third) , but some choices give the same permutations (for example $(12345)$ and $(23451)$ ). Such a choice gives 5 identical permutations (each is obtained by sliding each number of one place to the right, just to make sure you read me) thus you get $120/5=24$. In addition, note that those 24 elements constitute the conjugacy class of that cycle.

Answer 3

I believe both parts can be answered by use of the following "lemmas"; if $\sigma,\tau \in S_n$ are disjoint, then:

• $\sigma \tau = \tau \sigma$
• $\operatorname{o}(\sigma\tau) = \operatorname{lcm}(\operatorname{o}(\sigma),\operatorname{o}(\tau))$

where $\operatorname{o}(\sigma)$ gives the order of the permutation $\sigma$; the second allows us to see what the elements of order $5$ are simply the $5$-cycles, which was answered by Lano.

Your second question can then be answered (at least in part) by the first part of the "lemma" above.