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 $F/N$? $\endgroup$– HJRWSep 19, 2014 at 17:00
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$\begingroup$ @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 $A$. $\endgroup$– PabloSep 19, 2014 at 21:11
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$\begingroup$ Oh, right. I thought it was the centralizer. $\endgroup$– HJRWSep 20, 2014 at 5:49
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