MATH 421 521 UCI Section B Intro to Abstract Algebra Problems

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)