List of top Mathematics Questions on Homomorphisms

Let \( C_n \) denote a cyclic group having \( n \) elements. If there is a surjective group homomorphism from \( C_n \) to \( C_{30} \), then the total number of such distinct surjective homomorphisms is .
Which ONE of the following statements is FALSE?

Let \( G \) be a noncyclic group of order 4. Consider the statements I and II: 
I. There is NO injective (one-one) homomorphism from \( G \) to \( \mathbb{Z}_4 \) 
II. There is NO surjective (onto) homomorphism from \( \mathbb{Z}_4 \) to \( G \) 
 

Let \( G \) be a nonabelian group, \( y \in G \), and let the maps \( f, g, h \) from \( G \) to itself be defined by \[ f(x) = yxy^{-1}, \quad g(x) = x^{-1} \quad \text{and} \quad h = g \circ f \circ g. \] Then