Abstract Algebra Practice Problems

In this page, you can find a list of Abstract Algebra practice problems.

You need to submit the assignment based on the problems below.

Group Theory

Q1: Define semigroup and monoid with examples.

Q2: Is the set of natural numbers form a group under addition? Give reasons.

Q3: Let G = {2ⁿ: 𝑛 ∈ 𝑍}. Prove that the set G forms an abelian group with respect to the multiplication.

Q4: Define a binary operation on Q⁺ by

a$\circ$b = $\dfrac{ab}{2}$.

Find the inverse of 4.

Q5: What is (ab)^-1 in a group? Give proper reasons.

Q6: Define the order of an element of group.

1. Find the order of (14) (23) in S₄.
2. Let a∈G be an element of order 90. Find the order of a⁴⁰.

Q7: Prove that the order of each element in a finite group 𝐺 is a divisor of |𝐺|. 

Q8: Define a cyclic group. How many generators a cyclic group of order 60 has?

1. Prove that every subgroup of a cyclic group is cyclic.
2. Prove that every group of prime order is cyclic.

Q9: State and prove Lagrange’s theorem for finite groups. Show the converse is not true by giving an example.

Q10: Define a group homomorphism. What is the kernel of a group hoomorphism Φ: G → $G’$?

1. Prove kerΦ is a normal subgroup of G.
2. Show that Φ is injective if and only if ker Φ is trivial.

Q11: State and prove first isomorphism theorem. Using this theorem prove that there does not exist a group homomorphism from 𝐺 onto 𝐺$’$ where |𝐺| = 10 and |𝐺$’$| = 6.

Ring Theory

Q12: Define zero divisors in a ring. Find the zero divisors in the ring (Z₈, +, ⋅).

Q13: What is the characteristic of a ring? Write down the characteristics of the rings (Z, +, ⋅) and (Z₅, +, ⋅).

Q14: Give the definition of a unit in a ring. Show that $\overline{m}$ is a unit of the ring (Z_n, +, ⋅) then gcd(m, n) = 1.

Q15: What is integral domain? Give examples.

Q16: Define an ideal of a ring.Let 𝑅 be the ring of all continuous functions defined on [0,1]. Prove that 𝑆 = {𝑓∈𝑅: 𝑓($\frac{1}{2}$)= 0} is an ideal of 𝑅. 

Q17: Define a field. Prove that a field is an integral domain.

Q18: Define a ring homomorphism.