Homework Number 
Problem(s) 
# 2 
Let p be a prime and n a positive integer. Find a formula for the largest power of p which divides n!=n·(n1)···2·1. (Hint: For any real number x, we denote the largest integer less than or equal to x as [x] and call this the greatest integer function.) 
# 4 
Prove that a group of even order must have an element of order 2. 
# 5 
Let G be a finite group with more than one element. Show that G has an element of prime order. 
# 7 
If G_1, G_2, ..., G_r are all groups, show that Z(G_1)+Z(G_2)+...+Z(G_r)=Z(G_1+G_2+...+G_4) where + is the direct product "plus" and Z(G_i) is the center of the group G_i. (So, show the center of a direct product is the direct product of the centers.) Then use this to conclude that the direct product of a group is abelian if and only if each of the factors is abelian. 
# 10 
Suppose that N is a normal subgroup of a finite group G, and H is a subgroup of G with G/N prime. Prove that H is contained in N or that NH=G. 
# 11 
For every integer n>1, prove that (n1)! mod n =n1 if and only if n is prime. 

