**189-235A:** Basic Algebra I

## Assignment 8

## Due: Wednesday, November 5.

**1-2. ** Page 57, 2.4, 2.8.

**3-4. ** Page 66, 2.10, 2.11,

** 5-7. **Page 77, 2.27, 2.28, 2.29.

** 8. ** Let R be a ring.

a) Show that the set R endowed with the binary operation of addition is
a commutative group, with 0 as the identity element.

b) Show that the units in R endowed with the bonary operation of
multiplication is a group, with 1 as the identity element.

c) Let R now be a commutative ring, and let GL(n,R) be the group of units
in the ring M(n,R) of all n by n
matrices with entries in R (with multiplication
as the group operation). Show that GL(n,R) is a non-commutative group if
n >1.

d) Let p be a prime.
How many elements does the finite group GL(2,Z_p) contain? (Here Z_p denotes
the ring of integers modulo p.)

e) (Extra credit) How many elements does GL(3,Z_p) contain? How about
GL(n,Z_p) for arbitrary n?