> < ^ Date: Tue, 11 Dec 2001 11:51:08 -0800 (PST)
> < ^ From: Alireza Abdollahi <alireza_abdollahi@yahoo.com >
> ^ Subject: Answer to my question.

In the name of God
Dear Gap Forums,
Some days ago, I posed the following question in
GAP Forum:

 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 firstly thought that the answer is "No", and perhaps
we can produce an example with the aid of GAP, so I
decided for posing this question in the forum, but
now, I think
the answer is "yes". And I found it as follows:
One can see the following inequality:

$$1/|X| \sum_{x\in X} dim C_R(x) \leq 1/2 dim R$$

 Thus there is an element $x\in X$ such that 
$dim C_R(x) \leq 1/2 dim R$. On the other hand
$R=C_R(x) \times [R,x]$, thus $|C_R(x)|\leq
|[R,x]|\leq n$, so the result.

With best wishes for all you
Alireza Abdollahi

Alireza Abdollahi
Department of Mathematics
University of Isfahan,
Isfahan 81744,Iran
e-mail: alireza_abdollahi@yahoo.com
URL: http://www.abdollahi.8m.net

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