Prove that a cyclic group with more than one element may be a homomorphic image of a non cyclic group.

Question

Prove that a cyclic group with more than one element may be a homomorphic image of a non cyclic group. I know that I have to show that a surjective function exists from a non cyclic group to a cyclic group ($\mathbb{Z}$ or $\mathbb{Z}_n$) but I cant figure out any such function. Any hints on how to proceed especially what cyclic or non cyclic groups I can take for the function?

Answer 1

Consider the group $G=\mathbf{Z_{n}} \times \mathbf{Z_{n}}$. Consider thhe homomorphism $G \to \mathbf{Z_{n}}, (a,b) \to a$. G is not cyclic since gcd(n,n) is not 1. More generally, let H be a non trivial cyclic group, then $H \times H$ is non cyclic, then you have a homomorphism $H \times H \to H$, $(a,b) \to a$

Answer 2

Consider the sign homomorphism on the symmetric group, $sgn:S_n\to\{1,-1\}$.

Answer 3

How about $\varphi: V \rightarrow \mathbb{Z}_{2}$ where $V$ is Klein 4-group (non-cyclic) by $\varphi(e) = 0$ and $\varphi(a) = \varphi(b) = \varphi(c) = 1$