Let $G = S_3$ be the permutatiin group of 3 symbols.Then
- $G$ is isomorphic to a subgroup of a cyclic group
- There exists a cyclic group $H$ such that $G$ maps homomorphically onto $H$.
- $G$ is a product of cyclic groups
- there exists a nontrivial group homomorphism from $G$ to the additive group ($\mathbb Q $,+) of rational numbers
1 option is clearly false since subgroup of a cyclic group is again cyclic and $G$ can't be isomorphic to a cyclic group
2 option is true since there is an epimorphism from $G$ onto $\mathbb Z_2 $
3 option is false since $G$ is non commutative
Now I only get trivial homomorphism from $G$ to additive group of rational numbers. So option 4 is false. Am I right?