# 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 $66$.