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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.

eabcd

ee????

a?b??e

b?cde?

c?d?ab

d?????

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. LetGbeagroupandH asubgroupofG. LetC(H)={x∈G:xh=hxforallh∈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)

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)

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