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.

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Mid Term Practice Problems 2026

1. Define groupoid, semigroup and monoid with examples.
2. Is $(\mathbb{R}, .)$ a monoid?
3. Define a binary operation $\circ$ on $\mathbb{Z}$ $a \circ b=a+b-ab$ for all $a, b \in \mathbb{Z}$. Show $(\mathbb{Z}, \circ)$ is a monoid. What is the identity element?
4. Write down the definition of a group $(G, \circ)$. Prove that $(\mathbb{Z},+)$ is an abelian / commutative group.
5. Give reasons why these are not groups?: $(\mathbb{N}, -)$, $(\mathbb{Z}, .)$, $(\mathbb{R}, .)$, $(M_2(\mathbb{R}), .)$
6. Prove that a group always has only one identity element.
7. Prove that in a group each element has only one inverse.
8. In a group $(G, \circ)$, prove that $(a \circ b)^{-1}=b^{-1} \circ a^{-1}$ for all $a , b \in G$.
9. Prove that a group $(G, \circ)$ is abelian if and only if $(a \circ b)^{-1}=a^{-1} \circ b^{-1}$ for all $a , b \in G$.
10. Define a subgroup with examples. Write down subgroups of the groups $(\mathbb{Z},+)$, $(\mathbb{Q},+)$ and $(\mathbb{R},+)$.
11. Show that $SL_2(\mathbb{R})$ is a subgroup of $GL_2(\mathbb{R})$.
12. If $H$ and $K$ are two subgroups of a group $G$, then prove that $H\cap K$ is a subgroup of $G$.
13. Define the order an element $a$ in a group $(G, \circ)$. What is the order of $0$ in $(\mathbb{R},+)$?
14. Prove that both $a$ and $a^{-1}$ have the same order in a group.
15. If $\text{ord}(a)=n$ and $a^m=e$, then prove that $n$ divides $m$.
16. Let $a$ be an element of order $30$ in a group $(G, \circ)$. Find the order of $a^{18}$.
17. Find all elements of order $8$ in the group $(\mathbb{Z}_{24},+)$.
18. Find the of order $\overline{8}$ in the group $(\mathbb{Z}_{30},+)$.
19. Let f = (1 2 3) and g = ( 1 2) (3 4) in S₄. Find (fg)^-1.

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Group Theory 2025

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 2025

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.