# CBAS# SPMS# Dept. of Mathematics# Math 354 (Abstract Algebra I)
Problem Set 1: Hand in clearly typed/written solutions to all the problems except Problems
1(c), 1(d), and 1(e) by 17:30 on Friday, 22nd July 2022.
Problem Set 2: Hand in typed/written solutions. Problems 1(c), 1(d), and 1(e) are reading
problems. Study chapter 4, section 4.2 of Thomas W. Judson’s book on Abstract Algebra:
Theory and Applications to solve them. A copy of the book is already on Sakai. The deadline
for submission is 29th July 2022.
PLAGIARISM: You can discuss the problems with your colleagues, but solutions
must be written individually. If you used any source from the internet, please cite
it appropriately.
1. Find the number of elements in the following cyclic groups.
(a) The cyclic subgroup of Z30 generated by 25.
[5 Marks]
(b) The cyclic subgroup of Z42 generated by 30.
[5 Marks]
∗
(c) The cyclic subgroup ⟨i⟩ of the group C of nonzero complex numbers under multiplication.
[10 Marks]
(d) The cyclic subgroup of the group C∗ of nonzero complex numbers under multiplica√ .
tion generated by 1+i
2
[10 Marks]
(e) The cyclic subgroup of the group C∗ of nonzero complex numbers under multiplication gernerated by 1 + i.
[10 Marks]
2. Let G = ⟨x⟩ be a cyclic group of order 144. How many elements are there in the subgroup
⟨x26 ⟩?
[5 Marks]
3. List all the elements of Z45 that are of order 15.
4. Show that Zp has no proper nontrivial subgroups if p is a prime number.
[5 Marks]
[10 Marks]
5. Let G be a group and suppose a ∈ G generates a cyclic subgroup of order 2 and is the
unique such element. Show that ax = xa for all x ∈ G. [Hint: Consider (xax−1 )2 .]
[5 Marks]
6. Obtain all subgroups of Z18 and give their subgroup diagram.
1
[10 Marks]
[ALM & KD]
# CBAS# SPMS# Dept. of Mathematics# Math 354 (Abstract Algebra I)
7. Let G be an abelian group and let H and K be finite cyclic subgroups with |H| = r and
|K| = s.
(a) Show that if r and s are relatively prime, then G contains a cyclic subgroup of order
rs.
[5 Marks]
(b) Generalizing part (a), show that G contains a cyclic subgroup of order the least
common multiple of r and s.
[5 Marks]
8. If G is a group and a, b ∈ G, prove that ab and ba have the same order.
2
[10 Marks]
[ALM & KD]