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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?