10-1100/Homework Assignment 3

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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-p subgroup, for some prime p dividing 56.

Problem 2. (Qualifying exam, May 1997) Let S_5 act on ({\mathbb Z/5})^5 by permuting the factors, and let G be the semi-direct product of S_5 and ({\mathbb Z/5})^5.

  1. What is the order of G?
  2. How many Sylow-5 subgroups does G have? Write down one of them.

Problem 3. (Selick) Show that the group Q of unit quaternions (\{\pm 1, \pm i, \pm j, \pm k\}, subject to i^2=j^2=k^2=-1\in Z(Q) and ij=k) is not a semi-direct product of two of its proper subgroups.

Problem 4. (Qualifying exam, September 2008) Let G be a finite group and p be a prime. Show that if H is a p-subgroup of G, then (N_G(H):H) is congruent to (G:H) mod p. You may wish to study the action of H on G/H by multiplication on the left.

Problem 5. (easy)

  1. Prove that in any ring, (-1)^2=1.
  2. Prove that even in a ring without a unit, (-a)^2=a^2.

(Feel free to do the second part first and then to substitute a=1).

Problem 6.

  1. (Qualifying exam, April 2009) Prove that a finite integral domain is a field.
  2. (Qualifying exam, September 2008) Prove that in a finite commutative ring, every prime ideal is maximal.

Problem 7. (Dummit and Foote) A ring R is called a Boolean ring if a^2=a for all a\in R.

  1. Prove that every Boolean ring is commutative.
  2. Prove that the only Boolean ring that is also an integral domain is {\mathbb Z}/2.