2007 April

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.

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

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).

Short answers.
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 $G$.
b. Give an example of a simple group that is not cyclic.
c. Give an example of a solvable group that is not nilpotent.
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 $G$.
e. Make a list of abelian groups of order 72 such that every abelian group of order 72 is isomorphic to exactly one group on your list.
f. Give a solvable series for $S_4$, the symmetric group on four elements. 
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 $G$.