Suppose $G=V \rtimes M$, is a semi product group of an elementary abelian p-group of size $|V|=p^e$ and $M$ is a subgroup of $G$. If $f$ is the natural projection from $G$ onto $M$. $C_x=\{x^G\}$ is a conjugacy class. I would like to prove $|f^{-1}(m)\cap C_x|\geq p, m\in M$. Do you think such result is true? Best regards Ha.
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1$\begingroup$ Shouldn't you assume $e > 0$? $\endgroup$– S. Carnahan ♦Commented Jun 25, 2014 at 17:29
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In general it is false. Let $M$ be abelian and act trivially on $V$. Then $G$ is just the product of $M$ and $V$ so $G$ is commutative. Then $C_x$ contains only one element.
