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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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