# 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-${\displaystyle p}$ subgroup, for some prime ${\displaystyle p}$ dividing 56.

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

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

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

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

Problem 5. (easy)

1. Prove that in any ring, ${\displaystyle (-1)^{2}=1}$.
2. Prove that even in a ring without a unit, ${\displaystyle (-a)^{2}=a^{2}}$.

(Feel free to do the second part first and then to substitute ${\displaystyle a=1}$).

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 ${\displaystyle R}$ is called a Boolean ring if ${\displaystyle a^{2}=a}$ for all ${\displaystyle a\in R}$.

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