All homework are required to be typed in LaTeX. See the attached for more details.

MATH 421/521 Section B Intro to Abstract Algebra HW1 — Fall 2020

All homework are required to be typed in LaTeX. You can use the free online editor

http://www.overleaf.com. See https://www.overleaf.com/learn/ for a brief introduction.

HW 1 is due Thursday August 27 by 11:59pm. Please upload your solutions on canvas under

Assignments by the due time.

1. Consider the Dihedral group Dn that consists of n rotations and n reflections of a regular n-gon.

Assume that we already know Dn is a group (which means the composition of any two of the 2n

motion yields one of the original 2n motions.)

(a) Explain why a reflection followed by a reflection might be a rotation.

(b) Explain why a rotation followed by a reflection must be reflection.

(hint: a reflection always has a fixed vertex/corner)

2. For each positive integer n, the group of units U (n) contains positive integers that are less than

n and are relatively prime to n. The group operation is the multiplication modulo n.

(a) Write down the operation table (Cayley table) of U (18).

(b) Find the inverse of 7.

(c) Find the order of the element 7 in U (18).

3. (a) Show that if G is a group then in its operation table each element must appear exactly once

in each row and each column. (So you need to show that if a, b are any elements of G then there

is precisely one element c such that ac = b and there is precisely one element d such that da = b.

(b) Below is a partial operation table of a group G on {e, a, b, c, d}. Fill in the missing elements in

the table.

e

a

b

c

d

e

e

?

?

?

?

a

?

b

c

d

?

b

?

?

d

?

?

c

?

?

e

a

?

d

?

e

?

b

?

4. Let G be a group with the property that for that x, y, z in the group, xy = zx implies y = z.

Prove that G must be abelian.

5. Let G be a group and H a subgroup of G. Let C(H) = {x ∈ G : xh = hx for all h ∈ H}. C(H)

is called the centralizer of H in G. Prove that C(H) is a subgroup of G.

6. (Graduates only) Let G be a finite group. Prove that the number of nonidentity elements x of

G that satisfy x5 = e is a multiple of 4. (Don’t use any knowledge beyond what we covered so far.

However, you are allowed to the use the following fact that follows from the Euclidean algorithm:

if a, b are two relatively prime integers, then there exist integers x, y such that ax + by = 1)