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Let $p$ be a prime number, $F$ a free pro-$p$ group, $H \leq_c F$ of infinite index. Can it be that $$\sup_{N \lhd_o F} |Z((F/N)/C_{F/N}(HN/N))| < \infty ?$$

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  • $\begingroup$ Why is $C_{F/N}(HN/N)$ normal in $F/N$? $\endgroup$
    – HJRW
    Sep 19, 2014 at 17:00
  • $\begingroup$ @HJRW: For groups $A \leq B$ I write $C_{B}(A)$ for the normal core of $A$ in $B$ - the intersection of all conjugates of $A$ in $B$. This is the largest normal subgroup of $B$ contained in $A$. $\endgroup$
    – Pablo
    Sep 19, 2014 at 21:11
  • $\begingroup$ Oh, right. I thought it was the centralizer. $\endgroup$
    – HJRW
    Sep 20, 2014 at 5:49

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