Let $G$ be a non-abelian group. Assume that for every two non-central elements $x$ and $y$ there is a non-central element $w \in C_G(x) \cap C_G(y)$. Then what can we say about $G$?

Do you ask this because of your interest in the commuting graphs of groups? There seems to be no (finite?) group known to have this property - see the last sentence here.
– SomeoneApr 5 '13 at 13:41

1

@Someone: One of the comments on that same blog post mentions that there is a group of order 32 whose commuting graph has diameter 2.
– S. Carnahan♦Apr 5 '13 at 14:40

@S. Carnahan: Oh, right. Thanks. Even the very first comment corrects already the statement that no such groups are known. It seems worth to read the comments sometimes...
– SomeoneApr 5 '13 at 15:28

commuting graphsof groups? There seems to be no (finite?) group known to have this property - see the last sentence here. – Someone Apr 5 '13 at 13:41