WJD Open Notebook

2004 November

1. a. Carefully state (without proof) the three Sylow theorems.
   b. Prove that every group of order $p^2 q$, where $p$ and $q$ are primes with $p< q$ and $p$ does not divide $q-1$, is abelian.
   c. Prove that every group of order 12 has a normal Sylow subgroup.

2. a. Among finite groups, define nilpotent group, in terms of a particular kind of normal series. State two conditions on $G$ which are equivalent to the condition that $G$ is nilpotent.
   b. Prove that the center $Z(G)$ of a nilpotent group is non-trivial.
   c. Give an example which shows that $N \triangleleft G$ with both $N$ and $G/N$ nilpotent is not sufficient for $G$ to be nilpotent.

3. a. If $K, L$ are normal subgroups of $G$ prove that $G/K\cap L$ is isomorphic to a subgroup of $G/K \times G/L$ (the external direct product). What is the index of this subgroup in $G/K \times G/L$, in terms of $[G:K], [G:L]$ and $[G:KL]$?
   b. Prove either direction of: If $N\triangleleft G$, then $G$ is solvable if and only if both $N$ and $G/N$ are solvable.

4. a. If $H$ is a subgroup of $G$ and $[G:H]=n \geq 2$, prove that there exists a homomorphism $\rho$ from $G$ into $S_n$, the group of all permutations of an $n$-element set. Show that the kernel of $\rho$ is contained in $H$, and the image of $\rho$ is a transitive subgroup of $S_n$.
   b. If $[G:H]=n$ and $G$ is simple, then $G$ is isomorphic to a subgroup of $A_n$, the subgroup of all even permutations.
   c. Every group of order $2^3 \cdot 3^2 \cdot 11^2$ is solvable.

5. Let $G$ be a finite group, and suppose the automorphism group of $G$, ${{\mathrm{Aut}}}(G)$, acts transitively on $G\setminus {1}$. That is, whenever $x, y \in G\setminus {1}$ there exists an automorphism $\alpha\in {{\mathrm{Aut}}}(G)$ such that $\alpha(x) = y$. Prove that $G$ is an elementary abelian $p$-group for some prime $p$, by proving that:
   a. All non-identity elements of $G$ have order $p$, for some prime $p$.
   b. Here $Z(G) \neq 1$, and the center of every group is a characteristic subgroup.
   c. Thus $G\neq Z(G)$, i.e., $G$ is abelian.

6. Suppose $N$ is a normal subgroup of $G$. $C_G(N)$ denotes ${g\in G \mid g^{-1}ng = n \text{ for each } n\in N}$.
   a. Prove that $C_G(N)\triangleleft G$, and if $C_G(N) = {1}$ then $|G|$ divides $|N|!$.
   b. If $N$ is also cyclic, prove that $G/C_G(N)$ is abelian, and hence that $G' \leq C_G(N)$, where $G'$ is the commutator subgroup of $G$.