Math 520A Written Assignment No. 3

due Friday, March 30, 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. For each field $F$ find a subgroup $H$ of ${\mathrm{GL}}_n\left(F\right)$ such that ${\mathrm{GL}}_n\left(F\right)$ is isomorphic to the semi-direct product of $H$ with ${\mathrm{SL}}_n\left(F\right)$ for the action of the former (by conjugation within ${\mathrm{GL}}_n\left(F\right)$) on the latter.

2. Let $F$ be a field, and let $\alpha$ denote the action by fractional linear transformations of ${\mathrm{GL}}_2\left(F\right)$ on the set $X=F\cup \left\{\infty \right\}$.

   1. Show that $\alpha$ is a transitive action, i.e., there is only one orbit in $X$.

   2. What is the isotropy group at $\infty$?

   3. Explain briefly why $\alpha$ is induced by an action of ${\mathrm{PGL}}_2\left(F\right)$.

3. Let ${\mathbf{F}}_4=\mathbf{Z}⁄2\mathbf{Z}\left[t\right]⁄\left(t^2+t+1\right)\mathbf{Z}⁄2\mathbf{Z}\left[t\right]$ be the field with $4$ elements. Find the simple quotients for a composition series of ${\mathrm{SL}}_2\left({\mathbf{F}}_4\right)$.

4. Find the group of automorphisms of the quaternion group

5. One knows that the alternating group $A_5$ is a simple group of order $60$. Prove that any simple group of order $60$ must be isomorphic to $A_5$.