I have a problem that I have no idea how to begin solving. I need to describe the following group using an 'action table' (when the action is composition of homomorphisms). Moreover, I need to find out what is isomorphic to this group.

The group is : $\displaystyle Aut(GL_n(Z_7) / SL_n(Z_7))$ , which is a quotient group, with a given n>0, and knowing that $\displaystyle Z_7$ is a field.

Now, I believe I need to find $\displaystyle GL_n(Z_7) , SL_n(Z_7)$ , and find the order of each group. Then, I'll be able to know how many cosets the quotient group contains. Then, I'll have to actually FIND and describe the different automorphisms on this quotient group... Afterward, I'll need to check what happens whenever I composite each and every automorphism to another, and put it in a table, as I was asked to.

At last, I'll need to find what this group of automorphisms is isomorphic to...

Any help

??

Thanks a million!