$S_n, A_n, D_n, \ldots$

Let $\mathbb {F}_{2} = \{ 0,1\}$ be the field with two elements. Let $G$ be the group of invertible $2 \times 2$ matrices with entries in $\mathbb {F}_{2}$. Show that $G$ is isomorphic to $S_{3}$, the group of permutations of three objects.

Let $S_{n}$ denote the group of permutations of $n$ objects. Find four different subgroups of $S_{4}$ isomorphic to $S_{3}$ and nine isomorphic to $S_{2}$.

Let $G$ be a subgroup of $S_{5}$, the group of all permutations of five objects. Prove that if $G$ contains a 5-cycle and a 2-cycle, then $G = S_{5}$.

Let $G$ be a subgroup of the symmetric group on six objects, $S_{6}$. Assume that $G$ has an element of order 6. Prove that $G$ has a normal subgroup $H$ of index 2.

Let $S_{7}$ be the group of permutations of a set of seven objects. Find all $n$ such that some element of $S_{7}$ has order $n$.

$S_{9}$ is the group of permutations of 9 objects.

Let $S_{9}$ denote the group of permutations of nine objects and let $A_{9}$ be the subgroup consisting of all even permutations. Denote by $1 \in S_{9}$ the identity permutation. Determine the minimum of all positive integers $m$ such that every $\sigma \in S_{9}$ satisfies $\sigma^{m} = 1$. Determine also the minimum of all positive integers $m$ such that every $\sigma \in A_{9}$ satisfies $\sigma^{m} = 1$.

Let $S_{999}$ denote the group of permutations of 999 objects, and let $G \subset S_{999}$ be an abelian subgroup of order 1111. Prove that there exists $i \in \{ 1, \ldots , 999\}$ such that for all $\sigma \in G$, one has $\sigma (i) = i$.

Prove that $S_{n}$, the group of permutations of $\{ 1,2,\ldots ,n\}$, is isomorphic to a subgroup of $A_{n+2}$, the alternating subgroup of $S_{n+2}$.

Let $S_{n}$ be the group of all permutations of $n$ objects and let $G$ be a subgroup of $S_{n}$ of order $p^{k}$, where $p$ is a prime not dividing $n$. Show that $G$ has a fixed point; that is, one of the objects is left fixed by every element of $G$.

Let $G$ be a subgroup of $S_{n}$, the group of permutations of $n$ objects. Assume $G$ is transitive; that is, for any $x$ and $y$ in $S$, there is some $\sigma \in G$ with $\sigma (x) = y$.

Let $G$ be a transitive subgroup of the group $S_{n}$ of permutations of $n$ objects $\{ 1, \ldots , n\}$. Suppose that $G$ is a simple group and that $\sim$ is an equivalence relation on $\{ 1, \ldots , n\}$ such that $i \sim j$ implies that $\sigma (i) \sim \sigma (j)$ for all $\sigma \in G$. What can one conclude about the relation $\sim$?

Let $D_{n}$ be the dihedral group, the group of rigid motions of a regular $n$-gon ($n \geq 3$). (It is a noncommutative group of order $2n$.) Determine its center $Z = \{ c \in D_{n} \mid cx = xc \text{ for all } x \in D_{n}\}$.

How many Sylow 2-subgroups does the dihedral group $D_{n}$ of order $2n$ have, when $n$ is odd?