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(a) Find the number of p-Sylow subgroups in G = GL(n,F_{p}).

(b) For two p-Sylow subgroups H and J of G, consider the number of elements in the intersection. What number appears this way for given p and n?

(c) For a given p-Sylow subgroup H of G, find the number of p-Sylow subgroups with the given number of elements in the intersection with H.

2.

(a) Describe all groups of order 9.

(b) Describe all groups of order 27.

3. Find the table of irreducible characters for A_{4}.

4. Find the dimensions of irreducible representations of S_{4}.

5. For the non-trivial 1-dimensional irreducible representations of A_{4}, find the character of the induced representation of S_{4}, and determine if this representation is irreducible.

I have some idea how to do these questions (or at least some of them). I'll post some work a little later when I get time. But although I have some idea as to what to do, I don't really have any idea as to what's going on. I think right now I'm at the point that I'm trying to get comfortable with the definitions and basic results by applying them to actual problems, so I hope if someone can help me with the above problems, I can get a clearer picture of the underlying concepts.

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# Group Theory problems

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