Some group conditions imply non-trivial centre

During my research, I encountered the class of finite groups $$G$$ satisfying the following conditions:

1. $$G=\langle x_1,\dots, x_n\rangle$$
2. All the $$x_i$$ belong to the same conjugacy class of $$G$$ (hence they have the same order, say $$m$$)
3. $$\langle x_i \rangle \cap \langle x_j \rangle\neq 1$$ for all $$i,j$$.

My experiments using GAP indicate that a group $$G$$ with the properties described above has a non-trivial center. Therefore, I am currently working on proving this result which would be enough to prove what I need.

Remark: The statement is trivially true when $$n=2$$.

Approaches: I have verified the statement using brute-force computations and GAP's SmallGroups library for groups up to order 250. An inductive argument on the number of generators appears to be a promising approach, given the strength of the inductive hypothesis. However, I am having difficulty extending this argument to include the additional generator in the case where $$n>2$$.

I would appreciate any comments or suggestions on how to proceed with the proof. Thanks in advance.

• Why is the statement true for $n=2$? Given $x,y \in G$ and positive integers $m$ and $n$ with $G=\langle x, yxy^{-1} \rangle$ and $x^m=(yxy^{-1})^n \ne e$, it is not clear that $x=yxy^{-1}$. Commented Aug 7 at 17:26
• @GeoffreyTrang: The subgroup $\langle x_1\rangle\cap\langle x_2\rangle$ is centralized by both $x_1$ and $x_2$. Commented Aug 7 at 17:27

Here is a counterexample.

$$G = \langle x,y,z \rangle = S_{11}$$ with

$$x=(1,2)(3,4,5)(6,7,8,9,10)$$,

$$y = (1,2)(3,4,6)(7,8,9,10,11)$$,

$$z=(1,6)(3,4,5)(7,8,9,10,11)$$.

• +1. Just came to post (essentially) the same answer after playing around in GAP, but you beat me to it :) Commented Aug 7 at 17:24
• Thanks for your fast and correct counterexample. Unfortunately there is one. However, what if the group is solvable, in addition to the assumptions above? Does it have non-trivial center this time?
– fspa
Commented Aug 7 at 17:50
• @fspa That should probably be a separate question. Commented Aug 8 at 7:03