Math 8202        Homework 1

<p>Date due: February 1, 2010

<p>
Hand in only the starred questions. One of the other problems may show
up on the first quiz on February 8.

<p>
If you want me to do in class any of the questions I have given you,
do let me know (on or after the due date for the assignment). For
instance, if you want me to do on February 1 (8, respectively) or
later any of the questions below (not, respectively) assigned to be
handed in, please tell me.

<p>
<b>Section 3.4</b>: 2, 5, 11*, 12*

<p><b>Hint to Problem 11</b>: If G were finite, you could use the hint
to Problem 8 in the text, i.e., consider a minimal nontrivial normal
subgroup A of G contained in H. To do this problem in general,
consider the derived series of H, see Section 6.1.

<p>
<b>Section 3.5</b>: 3, 4, 7 (rigid motions = rotations), 9*, 10, 17

<p>
A. Prove that A<sub>4</sub> is the only subgroup of S<sub>4</sub>
having order 12.

<p>
B*. Make a complete list of all possible composition series for
S<sub>4</sub>. Show that there are no more composition series other
than the ones you have indicated. [You may use results from other
homework questions.]