Is a quotient group of a non-cyclic group again a non-cyclic group?

Question

Is a quotient group of a non-cyclic group again a non-cyclic group?

I know information about quotient groups of cyclic groups but don't know about non-cyclic

Answer 1

No.

Consider the Klein four group $$\langle a,b\mid a^2, b^2, ab=ba\rangle$$ and quotient out by the normal subgroup generated by $a$; that is, "kill $a$".

Answer 2

A metacyclic group $G$ is precisely an extension of a cyclic group $Q$ by a normal cyclic subgroup $N$, i.e. if there exists a short exact sequence $$1\rightarrow N \hookrightarrow G\twoheadrightarrow Q\to 1.$$ Such groups are not necessarily cyclic, nor even abelian.

For instance $S_3$ is metacyclic since its quotient by the subgroup generated by a $3$-cycle is cyclic. Dihedral groups are metacyclic too.