Simple Groups

Prove that a group of order 12 either has a normal subgroup of order 3, or a normal subgroup of order 4, or both.

Prove that a group of each of the following orders has a normal subgroup of order 5.

1. 15

2. 20

Consider the permutation $\sigma=\begin{pmatrix}1&3&2&4&5\end{pmatrix}$.

1. Explain why $\sigma\in\mathrm{A}_{5}$.

2. Find the order of the subgroup $\langle\sigma\rangle$.

3. Find another subgroup of $\mathrm{A}_{5}$ with order 5.

Let $S$ be a non-empty set and let $G\leq\mathrm{Sym}(S)$ act on $S$. Show that $\sigma\mathrm{Stab}_{G}(x)\sigma^{-1}=\mathrm{Stab}_{G}(\sigma(x))$ for any $\sigma\in G$ and $x\in S$.

Let $\sigma,\pi\in\mathrm{S}_{n}$. Suppose $\sigma$ has cycle decomposition

where $a_{1},a_{2},\dots,a_{k_{1}},b_{1},b_{2},\dots,b_{k_{2}},\dots$ are all distinct. Show that

that is, $\pi\sigma\pi^{-1}$ is obtained from $\sigma$ by replacing each entry $i$ by $\pi(i)$.