WJD Open Notebook

2002 November

1. Let $A$ be a finite abelian group. Let $n$ be the order of $A$ and let $m$ be the exponent of $A$ ($m$ is the least positive integer such that $ma = 0$ for all $a\in A$).
   a. Show that $m\mid n$ ($m$ is a factor of $n$).
   b. Show that $A$ is cyclic if and only if $m=n$.

2. Let $G$ be a finite group. Let $H\leq G$ be a subgroup.
   a. Show that the number of distinct conjugates of $H$ in $G$ is $|G|/|N(H)|$, where $N(H)$ is the normalizer of $H$ in $G$ and $| \cdot |$ indicates order.
   b. Show that $G = \bigcup_{g\in G} gH g^{-1}$ if and only if $H = G$.
   c. Deduce that $G$ is generated by a complete set of representatives of the conjugacy classes of elements of $G$.

3. Let $p, q$ and $r$ be distinct prime numbers.
   a. List, up to isomorphism, all abelian groups of order $p^4$.
   b. Up to isomorphism, how many abelian groups of order $p^4q^4 r^3$ are there?

4. Let $G$ be a nonsolvable group of least order among nonsolvable groups. Show that $G$ is simple.

5. Let $G$ be a group of order $p^n m$, where $p$ is a prime number, $n\geq 1$ and $p$ is not a factor of $m$. Let $r$ be the number of Sylow $p$-subgroups of $G$ and let $E = {K_1, \dots, K_r}$ be the set of Sylow $p$-subgroups of $G$. Let $N(K_i)$ be the normalizer of $K_i$ in $G$, and let $G$ act on $E$ by conjugation.
   a. Show that $\bigcap_{i=1}^r N(K_i)$ is a normal subgroup of $G$.
   b. Show that $|\bigcap_{i=1}^r N(K_i)|\geq p^n/(r-1)!$
   c. Deduce that a group of order 48 is not simple.

6. Show that a finite group $G$ is nilpotent if and only if $G$ is isomorphic to the product of its Sylow $p$-subgroups.