List of top Mathematics Questions on Group Theory

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?

Let \(G\) be a subgroup of \(GL_2(\mathbb{R})\) generated by 

Then the order of \(G\) is ...........

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