I have a group generated by 3 elements: let it be $G=\langle a,b,c\rangle$. The concrete form is given by some matrices, but never mind; my question is purely group theoretic.
Let us assume the generating elements are pairwise free(i.e., each of $\langle a,b\rangle$, $\langle b,c\rangle$, $\langle c,a\rangle$ is isomorphic to $F_2$). The question is: Can it be possible that there exists a subgroup $H\le G$ such that $\langle a\rangle \lneq H$ and $Z(H)=\langle a\rangle$?
The simplest case: if $G$ is itself free, then clearly it's impossible, since every subgroup of a free group is again free.