
Exam Prep Help
For the dihedral group($\displaystyle D_4 \mbox{or} \ D_8 $) with $\displaystyle \sigma = \begin{pmatrix} 1&2&3&4\\2&3&4&1 \end{pmatrix} \ \mbox{and} \ \tau = \begin{pmatrix} 1&2&3&4\\2&1&4&3 \end{pmatrix} \mbox{and} $ $\displaystyle \ D_4 = ${$\displaystyle {1 , \sigma , \sigma ^{2}, \sigma ^{3}, \tau , \sigma \tau , \sigma ^{2} \tau , \sigma ^{3} \tau }$}
(3) Find $\displaystyle < \sigma^{2} , \tau > \ \mbox{and} \ <\sigma^{3} , \sigma \tau > $?
(4) What are the left cosets of $\displaystyle <\tau>, <\sigma^{2}>, \mbox{and} \ <\sigma^{2}, \tau> $?
(5) Prove if $\displaystyle <\tau> \ \mbox{and/or} \ <\sigma^{2}> $ is normal.
(6) Construct the Cayley table of $\displaystyle g/<\tau> $
I've tried really hard to work these questions out. I worked out the first two questions (from the attached image above) but I'm having trouble with the four posted here. I really need to be shown how to do these otherwise I wont get it. Ive been working on these for about 6 hours and Im not getting anywhere. (Crying)

This link may be helpful
http://en.wikipedia.org/wiki/Examples_of_groups
$\displaystyle D_8 \mbox{ or } D_4$ is the example you are looking for

I've finished Q1's and 2 now, but I'm completely stuck on questions (3)  (6). My midterm is tomorrow and I think I'll really need to know how to do this.
Thanks to anyone who can help.