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Let all groups here be finite $p$--groups.

Given $K<H$, let $r(K,H)$ be the smallest $r$ such that there exists a chain of subgroups $K=L_0 \lhd L_1 \lhd \cdots \lhd L_r = H$, such that each $L_i/L_{i-1}$ is cyclic.

Question: Let $H$ be a subgroup of $G$. If $K_1$ and $K_2$ are subgroups of $H$ that are conjugate in $G$, does it follow that $r(K_1,H) = r(K_2,H)$?

As in the last question I posed, a minimal counterexample would have $G = \langle H,g\rangle$, and $H$ not normal in $G$.

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    $\begingroup$ Your title asks "can this happen?", while your body asks (essentially) "must this happen?" Could a better (as more specific) title be something like "Lengths of composition-type series for conjugate subgroups of finite $p$-groups"? Also, I added a link to what I suppose is the last question. $\endgroup$
    – LSpice
    Commented Jun 28, 2019 at 0:38

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It does not follow. Let $G=Z_4\wr Z_2$, or equivalently $$G=\langle a,b,t|a^4=b^4=[a,b]=t^2=1, a^t=b\rangle.$$ Let $H=\langle a,b^2\rangle$, $K_1=\langle a^2\rangle$, and $K_2=\langle b^2\rangle$. Then $K_1^t=K_2$, $r(K_1,H)=2$, and $r(K_2,H)=1$.

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    $\begingroup$ Thanks! (Oh man, I should have thought of that.) $\endgroup$
    – Nicholas Kuhn
    Commented Jun 28, 2019 at 1:02

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