Assignment 1 – Due Friday, September 12
Turn this in at the start of recitation on Friday, September 12.
Part 1
This part only uses what we covered in class on September 3. I recommend that you complete it before class
on September 8.
1. Read Chapter 1 and Sections 2.1–2.2 of Herstein.
2. Herstein, p. 35: #1
3. An affine function on the line is a function f : R → R of the form f (x) = mx + b, where m, b ∈ R. Let
A1 be the set of affine functions with m 6= 0, and define an operation ∗ on A1 by letting f ∗ g be the
function (f ∗ g)(x) = f (g(x)).
(a) Suppose that f (x) = 2x + 1 and g(x) = −3x + 2. Compute f ∗ g and g ∗ f .
(b) Show that this operation makes A1 into a group. What is the identity element? Is A1 an abelian
group?
Part 2
1. Read Sections 2.3–2.4 of Herstein.
2. Herstein, p. 35: #2, 3
3. List and describe all the symmetries of a 2 × 1 rectangle. Write a table giving the products of all of
these symmetries. Show that these symmetries form a group.
4. List all of the subgroups of the group of symmetries of the 2 × 1 rectangle.
5. If G is a group and a ∈ G, the centralizer Z(a) of a is the set
Z(a) = {g ∈ G | ga = ag}.
(a) If e ∈ G is the identity element, what is Z(e)?
(b) Show that for any a ∈ G, the set Z(a) is a subgroup of G.
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