Math 3500 Modern Algebra I

Spring 2011

Challenge Problems

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·(n-1)···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 (n-1)! mod n =n-1 if and only if n is prime.

Last Updated: April 19, 2011