Prove that a group of order 56 has a normal Sylow p-subgroup for some prime p dividing its order

Prove that a group of order 56 has a normal Sylow p-subgroup for some prime p dividing its order

Irreducible Polynomial/Galois Group

Please see the attached PDF file. I would prefer a PDF format solution. Thanks much!

Describe the Group

Let G be a finite nonabelian group of order 27 where all the elements have order 3. Prove that there is exactly one such group G and give a complete description. Please explain your reasoning and solution in as much detail as possible. Thank You. (Question is also included in attachment) Please, this posting is reserved for Alexandru Ghitza OTA#103197, who has worked out a solution for this problem, and will post it here.

subset/subgroup problem

If G is a group and gEG, define C(g)={zEG|zg=gz} show that C(g) is a subgroup of G (the centralizer of g in G). Note: E means element of

Cyclic group problem

Let G be a cyclic group of order n. Show that g^n=1 for all gEG. If g^m=1 in G where gcd(m,n)=1, show that g=1