2008 January
Instructions. Do as many problems as you can. You may use earlier parts of a problem to solve later parts, even if you cannot solve the earlier part; however, complete solutions are preferred. Most importantly, give careful solutions.

Prove that if $H$ is a subgroup of a finite group $G$, then $H$ divides $G$ and use this to show that the order of each element of a finite group divides the order of the group.

a. State Sylow’s theorems.
b. Show that there is no simple group of order 56.
c. Show that every group of order 35 has a normal subgroup of order 5 and one of order 7.
d. Is every group of order 35 abelian? 
How many nilpotent groups of order 360 are there? Justify your answer. Your justification should indicate clearly which theorems you are applying. If you are not able to find the number of nilpotent groups, at least find the number of Abelian groups of order 360.

Show that every group of order 1000 is solvable. You can use the fact that if $G$ has a normal subgroup $N$ such that both $N$ and $G/N$ are solvable, then $G$ is solvable.