WJD Open Notebook

2006 April

Instructions. Do as many problems as you can. You are not expected to do all of the problems. You may use earlier parts of a problem to solve later parts, even if you cannot solve the earlier part; however, complete solutions are preferred. Most importantly, give careful solutions.

1. Show that there is no simple group of order 992.

2. Let $G$ be a nonabelian simple group. Let $S_n$ be the symmetric group of all permutations on an $n$-element set, and let $A_n$ be the alternating group.
   a. Show that if $G$ is a subgroup of $S_n$, $n$ finite, then $G$ is a subgroup of $A_n$.
   b. Let $H$ be a proper subgroup of $G$, and, for $g\in G$, let $\lambda_g$ be the map of the set of left cosets of $H$ onto themselves defined by $\lambda_g(xH) = gxH$. Show that the map $g\mapsto \lambda_g$ is a monomorphism (injective homomorphism) of $G$ into the group of permutations of the set of left cosets of $H$.
   c. Let $H$ be a subgroup of $G$ of finite index $n$ and assume $n>1$ (so $H \neq G$). Show that $G$ can be embedded in $A_n$.
   d. If $G$ is infinite, it has no proper subgroup of finite index.
   e. There is no simple group of order 112.

3. a. Let $\alpha$ be an element of the symmetric group $S_n$ and let $(i_1 i_2 \dots i_k)$ be a cycle in $S_n$. Prove that $\alpha^{-1}(i_1 i_2 \dots i_k)\alpha= (i_1\alpha i_2 \alpha \dots i_k\alpha)$. (Note that it is assumed that permutations act on the right, so $\alpha$ maps $i$ to $i\alpha$.)
   b. Show that $A_4$ is not simple.
   c. Show that any five-cycle $\sigma \in S_5$ and any two-cycle $\tau \in S_5$ together generate $S_5$.

4. If $A$ and $B$ are subgroups of a group $G$, let $A\vee B$ be the smallest subgroup containing both, and let $AB = {ab : a\in A, b\in B}$.
   a. Show that if $A$ is a normal subgroup then $AB = A\vee B$ and that, if both $A$ and $B$ are normal, then $A\vee B$ is normal.
   b. If $A, B$ and $C$ are normal subgroups of $G$ and $C\subseteq A$, prove Dedekind's modular law: $A \wedge (B\vee C) = (A\wedge B) \vee C$.

5. Let $G$ be a group and let $Z$ be its center.
   a. Show that if $G/Z$ is cyclic then $G$ is abelian.
   b. Show that any group of order $p^2$, where $p$ is a prime, is abelian.
   c. Give an example of a non-abelian group $G$ where $G/Z$ is abelian.
   d. Let $\varphi$ be a homomorphism from $G$ onto $K$, where $K$ is an abelian group. Let $N$ be the kernel of $\varphi$ and suppose $N$ is contained in $Z$. Suppose that there is an abelian subgroup $H$ of $G$ such that $\varphi(H) = K$. Show $G$ is abelian.

6. Let $G$ be a finite group and let $H \triangleleft G$ be a normal subgroup. Show that $G$ is solvable if and only if $H$ and $G/H$ are solvable.