Math 520A Written Assignment No. 2

due Monday, March 12, 2007

Directions. This assignment should be typeset. You must explain the reasoning underlying your answers. If you make use of a reference other than class notes, you must properly cite its use.

You may not seek help from others on this assignment.

1. If $G$ is any group, $H$ a subgroup, and $x,y$ elements of $G$, prove that $xH=Hy$ if and only if $x$ and $y$ both belong to the normalizer $N_G\left(H\right)$ and determine the same element of $N_G\left(H\right)⁄H$.

2. For each relevant prime $p$ determine the number of Sylow $p$-subgroups of the symmetric group $S_5$.

3. Show that the group ${\mathrm{SL}}_2\left({\mathbf{F}}_3\right)$ has a normal subgroup of order $8$. List the $8$ elements of this subgroup, and explain why this group of order $8$ cannot be isomorphic to the dihedral group $D_4$.

4. Let $G$ be a finite group and $H$ a subgroup of index $3$ in $G$ that is not a normal subgroup of $G$. Show that $H$ contains a subgroup $N$ that is normal in $G$ for which $G⁄N\cong S_3$.

5. Find an explicit list of groups that represent all isomorphism classes of groups of order $66$.