MATH 128A University of California Santa Cruz Abstract Algebra Questions

Math 128A, Fall 2021, homework 2Due: midnight, Monday, September 13
Note: submit your work on Canvas
Problems to be done, but not turned in:
• Chapter 3: all odd numbers 1 – 39
Problems to be turned in:
1. Let

G=
a b
−b a

2
2

a + b = 1, a, b ∈ R
It can be shown that G is a group under matrix multiplication (i.e., you do not have
to prove this).
Is G an abelian group? Prove or disprove.
2. (Ch. 3) 18. Suppose that a is a group element and a6 = e. What are the possibilities
for |a| (the order of a)? Provide reasons for your answer.






0 −1
0 1
1 1
in GL(2, R). Find the order
, and C =
,B=
3. Consider A =
1 1
−1 0
0 1
of A, B and C.
4. If G is a group, a, b ∈ G, and both a and b have finite order, is it always the case that
ab has finite order? Prove or give a counterexample.
5. Show that U (14) = h3i = h5i. [Hence, U (14) is cyclic.] Is U (14) = h11i?
6. List the elements of the subgroups h3i and h15i in Z18 . Let a be a group element of
order 18. List the elements of the subgroups ha3 i and ha15 i.
7. Suppose H is a subgroup of Z under addition, and H contains 18 and 45. What are
the possibilities for H? Prove your answer.