# Homework 11

```Homework 11
Math 332, Spring 2013
These problems must be written up in LATEX, and are due this Friday, May 10.
1. Let G = U (5) &times; U (5), and let N = {(1, 1), (2, 3), (3, 2), (4, 4)}.
(a) List the elements of the four cosets of N in G.
(b) Determine the isomorphism type of G/N .
2. Let G = GL(2, R), and let N be the following normal subgroup of G:
a 0 N =
a ∈ R and a 6= 0 .
0 a 1 −1
(a) Determine the order of
N in G/N .
1 1
2 1
(b) Find an element A ∈ SL(2, R) such that AN =
N.
5 7
3. Let G = Z8 &times; Z2 .
(a) Find a four-element subgroup H of G such that G/H is isomorphic to Z2 &times; Z2 .
(b) Find a four-element subgroup K of G such that G/K is isomorphic to Z4 .
4. Let R be the group of real numbers under addition, let Z be the group of integers
under addition, and consider the quotient group R/Z.
(a) What is the order of 2/5 + Z in R/Z? Explain.
(b) The group R/Z has four elements of order eight. List them.
(c) Give an example of an element of R/Z with infinite order.
5. Let G be a group, let H be a subgroup of G, and let N be a normal subgroup of G.
Define
N H = {nh | n ∈ N and h ∈ H}.
(a) Prove that N H is a subgroup of G.
(b) Prove that if H is normal in G, then N H is also normal in G.
```