Dear Alireza Abdollahi, dear GAP Forum,

Last Friday you wrote to the GAP Forum:

I have encountered to the following question:

Let $n>3$ be a postive integer and let $G=RX$ be an
extention of an elementary abelian $p$-group by an
abelian $p'$-subgroup $X$ such that $X$ acts
faithfully on $R$ and $R=[R,X]$ and $|[R,x]|\leq n$
for all $x\in X$. It can be proved that $|X|\leq n-1$
and  $|R|\leq n^{\log_2(n-2)}$.
 Is it true that $|R|\leq n^2$?

I am sorry that I have no idea how to answer your question, but since
nobody from the GAP Forum has reacted as yet, the colleagues reading
the GAP Forum seem not to have an answer either.

However for a theoretical question such as the one you ask, the GAP
Forum is perhaps not the right address. The GAP Forum is meant for
discussing questions related to the use of GAP, but in the case of
your question the only chance to use GAP would be if somebody had an
idea for a possible counterexample and would then use GAP to verify
it.

Rather I want to suggest to send your question - as well as similarly
'theoretical' questions - to the 'group-pub-forum' maintained by Geoff
Smith at Bath, UK, that is read by many group theoreticians, not only
those who are interested in GAP.

With kind regards Joachim Neubueser