A finite, non-abelian group $G$ is said to be a center commutative-transitive group $($or a CCT-group, for short$)$ if commutativity is a transitive relation on the set on non-central elements. In other words, if $x, \, y, \, z \in G-Z(G)$ and $[x, \, y]=[y, \, z]=1$, then $[x, \, z]=1$.
A quick search with GAP4 allows to prove the following
Proposition. Let $G$ be a non-abelian finite group with $|G| <32$; then $G$ is a CCT-group, unless it is isomorphic to $S_4$. In the case $|G|=32$, there are precisely seven groups that are not CCT: the two extra-special groups (whose nilpotency class is $2$), and five further groups having nilpotency class $3$.
I tried to give a computer-independent proof, but it turned very soon into a lenghty and messy case-by case analysis. So let me ask the following
Question.
- Is there any short conceptual proof of the proposition above?
- Does the result in the proposition above appear somewhere in the literature?