List of top Mathematics Questions on Group Theory asked in IIT JAM MA

Let \( G \) be the power set of \( \{1, 2, 3, 4\} \). Let the binary operation \( \Delta \) on \( G \) be

Which of the following statements are false?

The number of group homomorphisms from the group \(\mathbb{Z}_4\) to the group \(S_3\) is _________.

The number of elements of order two in the group \(S_4\) is equal to _________.

The number of cycles of length 4 in \(S_6\) is _________.

Let \(G\) be a finite group of order \(28.\) Assume that \(G\) contains a subgroup of order \(7.\) Which of the following statements is/are true?

Let \( g \) be an element of \( S_7 \) such that \(g\) commutes with the element \((2,6,4,3)\). The number of such \(g\) is

Let \(G\) be a finite abelian group of odd order. Consider the following two statements: I. The map \( f : G \to G \) defined by \( f(g) = g^2 \) is a group isomorphism.
II. The product \( \displaystyle \prod_{g \in G} g = e. \)
Which one of the following statements is true?

Which one of the following statements is true?

Consider the following statements. I. The group \((\mathbb{Q}, +)\) has no proper subgroup of finite index.
II. The group \((\mathbb{C} \setminus \{0\}, \cdot)\) has no proper subgroup of finite index.
Which one of the following statements is true?

Which one of the following statements is true?

How many elements of the group \(\mathbb{Z}_{50}\) have order 10?

Let \[ G = \{ n \in \mathbb{N} : n \leq 55, \, \gcd(n, 55) = 1 \} \] be the group under multiplication modulo 55. Let \( x \in G \) be such that \( x^2 = 26 \) and \( x>30 \). Then \( x \) is equal to ................

Let \( H \) and \( K \) be subgroups of \( \mathbb{Z}_{144} \). If the order of \( H \) is 24 and the order of \( K \) is 36, then the order of the subgroup \( H \cap K \) is

Let \( \mathbb{C}^* = \mathbb{C} \setminus \{0\} \) denote the group of non-zero complex numbers under multiplication. Suppose \[ Y_n = \{ z \in \mathbb{C} \mid z^n = 1 \}, n \in \mathbb{N}. \] Which of the following is (are) subgroup(s) of \( \mathbb{C}^* \)?

Consider the group \( \mathbb{Z}^2 = \{ (a, b) | a, b \in \mathbb{Z} \} \) under component-wise addition. Then which of the following is a subgroup of \( \mathbb{Z}^2 \)?

Let \( G \) be a group satisfying the property that \( f: G \to \mathbb{Z}_{221} \) is a homomorphism implies \( f(g) = 0 \), \( \forall g \in G \). Then a possible group \( G \) is

Which one of the following is TRUE?

The order of the element \( (1 \ 2 \ 3)(2 \ 4 \ 5)(4 \ 5 \ 6) \) in the group \( S_6 \) is ..............

Let \( A_6 \) be the group of even permutations of 6 distinct symbols. Then the number of elements of order 6 in \( A_6 \) is ..................

Let \( H \) be the quotient group \( \mathbb{Q} / \mathbb{Z} \). Consider the following statements.
I. Every cyclic subgroup of \( H \) is finite.
II. Every finite cyclic group is isomorphic to a subgroup of \( H \).
Which one of the following holds?

The maximum order of a permutation \( \sigma \) in the symmetric group \( S_{10} \) is ..........

The number of generators of the additive group \( \mathbb{Z}_{36} \) is equal to