189-235A: Basic Algebra I
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
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?