Since time is short, I am going to discuss a couple of Groups questions but in slightly less detail than I have been giving up to now: instead of working through the questions completely, I’ll try to zero in on the most important points. Because there wasn’t a separate Groups course until 2008, I am taking my questions from that year.
5E. For a normal subgroup of a group , explain carefully how to make the set of (left) cosets of into a group.
For a subgroup of a group , show that the following are equivalent:
(i) is a normal subgroup of ;
(ii) there exist a group and a homomorphism such that is the kernel of .
Let be a finite group that has a proper subgroup of index (in other words, ). Show that if , then cannot be simple. [Hint: Let act on the set of left cosets of by left multiplication.]