10-1100/Homework Assignment 3
This assignment is due at class time on Tuesday, November 16, 2010.
Solve the following questions
Problem 1. (Selick) Show that any group of order 56 has a normal Sylow- subgroup, for some prime dividing 56.
Problem 2. (Qualifying exam, May 1997) Let act on by permuting the factors, and let be the semi-direct product of and .
- What is the order of ?
- How many Sylow-5 subgroups does have? Write down one of them.
Problem 3. (Selick) Show that the group of unit quaternions (, subject to and ) is not a semi-direct product of two of its proper subgroups.
Problem 4. (Qualifying exam, September 2008) Let be a finite group and be a prime. Show that if is a -subgroup of , then is congruent to mod . You may wish to study the action of on by multiplication on the left.
Problem 5. (easy)
- Prove that in any ring, .
- Prove that even in a ring without a unit, .
(Feel free to do the second part first and then to substitute ).
- (Qualifying exam, April 2009) Prove that a finite integral domain is a field.
- (Qualifying exam, September 2008) Prove that in a finite commutative ring, every prime ideal is maximal.
Problem 7. (Dummit and Foote) A ring is called a Boolean ring if for all .
- Prove that every Boolean ring is commutative.
- Prove that the only Boolean ring that is also an integral domain is .