[latex] \[
\textbf{Difficult MCQs on Group Theory and Applications with Answers}
\]
\[
\textbf{Q1: Which of the following is a group under matrix multiplication?}
\]
\[
\text{(A) The set of all \(2 \times 2\) invertible matrices}
\]
\[
\text{(B) The set of all \(2 \times 2\) matrices}
\]
\[
\text{(C) The set of all \(2 \times 2\) matrices with real entries}
\]
\[
\text{(D) The set of all \(n \times n\) matrices with integer entries}
\]
\[
\textbf{Answer: (A) The set of all \(2 \times 2\) invertible matrices}
\]
\[
\textbf{Q2: Which of the following is NOT a subgroup of the symmetric group \(S_3\)?}
\]
\[
\text{(A) The set of even permutations}
\]
\[
\text{(B) The set of odd permutations}
\]
\[
\text{(C) The set of all permutations of three elements}
\]
\[
\text{(D) The set containing the identity element and a single transposition}
\]
\[
\textbf{Answer: (B) The set of odd permutations}
\]
\[
\textbf{Q3: In a finite group, the order of any element divides the order of the group. This statement is known as:}
\]
\[
\text{(A) Lagrange’s Theorem}
\]
\[
\text{(B) Cauchy’s Theorem}
\]
\[
\text{(C) Sylow’s Theorem}
\]
\[
\text{(D) Fermat’s Little Theorem}
\]
\[
\textbf{Answer: (A) Lagrange’s Theorem}
\]
\[
\textbf{Q4: Which of the following is a property of an Abelian group?}
\]
\[
\text{(A) The group has only one element}
\]
\[
\text{(B) The group operation is commutative}
\]
\[
\text{(C) The group is always finite}
\]
\[
\text{(D) The group has no identity element}
\]
\[
\textbf{Answer: (B) The group operation is commutative}
\]
\[
\textbf{Q5: The direct product of two groups \(G\) and \(H\), denoted \(G \times H\), has the following property:}
\]
\[
\text{(A) The order of the product is the product of the orders of \(G\) and \(H\)}
\]
\[
\text{(B) The product has the same order as \(G\) or \(H\)}
\]
\[
\text{(C) The product is a subgroup of \(G\)}
\]
\[
\text{(D) The product is Abelian if and only if both \(G\) and \(H\) are Abelian}
\]
\[
\textbf{Answer: (A) The order of the product is the product of the orders of \(G\) and \(H\)}
\]
\[
\textbf{Q6: The group \( \mathbb{Z}_6 \) under addition modulo 6 is:}
\]
\[
\text{(A) Abelian and cyclic}
\]
\[
\text{(B) Non-Abelian and cyclic}
\]
\[
\text{(C) Non-Abelian and not cyclic}
\]
\[
\text{(D) Abelian and not cyclic}
\]
\[
\textbf{Answer: (A) Abelian and cyclic}
\]
\[
\textbf{Q7: Which of the following is true for a subgroup \( H \) of a group \( G \)?}
\]
\[
\text{(A) Every subgroup of a finite group is normal}
\]
\[
\text{(B) If \( G \) is Abelian, then every subgroup of \( G \) is normal}
\]
\[
\text{(C) Every subgroup of a cyclic group is cyclic}
\]
\[
\text{(D) Every subgroup of a non-commutative group is non-commutative}
\]
\[
\textbf{Answer: (C) Every subgroup of a cyclic group is cyclic}
\]
\[
\textbf{Q8: If \( G \) is a group and \( a \) is an element of \( G \), the order of \( a \) is the:}
\]
\[
\text{(A) The smallest integer \( n \) such that \( a^n = e \), where \( e \) is the identity element}
\]
\[
\text{(B) The largest integer \( n \) such that \( a^n = e \), where \( e \) is the identity element}
\]
\[
\text{(C) The number of distinct elements in \( G \)}
\]
\[
\text{(D) The number of elements in the cyclic subgroup generated by \( a \)}
\]
\[
\textbf{Answer: (A) The smallest integer \( n \) such that \( a^n = e \), where \( e \) is the identity element}
\]
\[
\textbf{Q9: Which of the following is true about a non-trivial normal subgroup?}
\]
\[
\text{(A) It is always cyclic}
\]
\[
\text{(B) It must be Abelian}
\]
\[
\text{(C) It is closed under the group operation}
\]
\[
\text{(D) It is always invariant under conjugation by elements of the group}
\]
\[
\textbf{Answer: (D) It is always invariant under conjugation by elements of the group}
\]
\[
\textbf{Q10: The alternating group \(A_n\) consists of all the even permutations of \( n \) elements. What is the order of \(A_5\)?}
\]
\[
\text{(A) 60}
\]
\[
\text{(B) 120}
\]
\[
\text{(C) 240}
\]
\[
\text{(D) 720}
\]
\[
\textbf{Answer: (A) 60}
\]