Abstract Algebra MCQ 2025 - 2026

Which of the following is not a property of a group?

• a) Closure

• b) Associativity

• c) Commutativity

• d) Existence of inverse

A group G is called Abelian if:

• a) ab=ba for all a,b∈Ga, b \in Ga,b∈G

• b) ab≠baab \neq baab=ba for all a,b∈Ga, b \in Ga,b∈G

• 
  c) G is finite

• d) G has no identity element

In a group G, the identity element is:

• a) Unique

• b) Not unique

• c) May not exist

• d) Equal to inverse of every element

The set of all 2×2 invertible matrices under multiplication forms:

• a) A group

• b) A ring

• c) A field

• d) None

The symmetric group S3​ has how many elements?

• a) 3

• b) 4

• c) 5

• d) 6

If H is a non-empty subset of a group G, then H is a subgroup of G if:

• a) a,b∈H⇒ab∈Ha, b \in H \Rightarrow ab \in Ha,b∈H⇒ab∈H

• b) a∈H⇒a−1∈Ha \in H \Rightarrow a^{-1} \in Ha∈H⇒a−1∈H

• c) H contains the identity element

• d) All the above

Every subgroup of a cyclic group is:

• a) Non-Abelian

• b) Cyclic

• c) Finite

• d) Normal

The subgroup generated by an element a of G is denoted by:

• a) (a)

• b) ⟨a⟩

• c) G(a)

• d) {a}

A subgroup H of G is called normal if:

• a) aH=Ha for all a∈G

• b) aH=e

• c) H=G

• d) H⊂G

If H is a subgroup of a group G, then for any a∈G, the set aH={ah:h∈H}

• a) Left coset of H in G

• b) Right coset of H in G

• c) Normal subgroup

• d) Quotient group