## 2007 April

1. Let $H$ and $K$ be (not necessarily normal) subgroups of a group $G$. For $g$ an element of $G$ the set $H g K$ is called a double coset. Show that any two double cosets are either identical or do not intersect.

2. State (without proof) the three Sylow theorems. Show that every group of order 56 contains a proper normal subgroup.

3. Define $G'$, the commutator subgroup of $G$. Show that $G'$ is a normal subgroup of $G$. Show that every homomorphism from $G$ to an abelian group $A$ factors through $G/G'$ (i.e., given $\varphi : G \rightarrow A$ there exists $\bar{\varphi} : G/G' \rightarrow A$ such that $\varphi = \bar{\varphi}\circ \pi$ where $\pi : G \rightarrow G/G'$ is natural).

a. Give examples of groups $K, N$, and $G$ with $K$ a normal subgroup of $N$, $N$ a normal subgroup of $G$, but $K$ not a normal subgroup of $G$.
d. Give an example of a group $G$ that is not abelian but $G/C(G)$ is abelian where $C(G)$ denotes the center of $G$.
f. Give a solvable series for $S_4$, the symmetric group on four elements.
5. Let $H$ be a finite index subgroup of $G$. Show there is a normal subgroup of $G$ that is contained in $H$ and has finite index in $G$.