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$G$ is a finite solvable group. Let $\{P_{1}, P_{2}, \dotsc , P_{s}\}$ be a Sylow basis of $G$. We have that $G=P_{1}P_{2}\dotsm P_{s}$. Set \begin{equation} \begin{aligned} %% The alignment is never used …. T=\prod\limits_{t=1}^{s-1}P_t, H=\prod\limits_{k\neq3}^sP_k, K=\prod\limits_{r\neq2}^sP_r.\nonumber \end{aligned} \end{equation} Suppose that $T$ is nilpotent (i.e. $T=P_{1}\times P_{2}\times P_{3}\times \dotsb \times P_{s-1}$), $N_H(P_s)=P_s$ and $N_K(P_s)=P_s$.

Can we get that $N_G(P_s)=P_s$?

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    $\begingroup$ This seems to be the same as your earlier question—which, as was observed there, you also posted on MSE—just with more primes. $\endgroup$
    – LSpice
    Nov 26, 2022 at 2:42
  • $\begingroup$ This question has more of a condition for $T$ to be nilpotent than the earlier question. $\endgroup$ Nov 26, 2022 at 2:45
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    $\begingroup$ OK. At least to me, it is not clear why these are natural conditions to consider. What is the motivation? $\endgroup$
    – LSpice
    Nov 26, 2022 at 3:45

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Yes. Nilpotency of $T$ is not needed. Assume that $N_G(P_s)>P_s$. Then for some $i<s$, $N_G(P_s)$ has a Sylow $p_i$-subgroup $Q_i\ne 1$. Assume without loss that $p_i$ divides $|H|$. Then $P_iP_s$ is a Hall $\{p_i,p_s\}$-subgroup of $H$ and $G$. By Hall's extension of Sylow's theorems to solvable groups, $(Q_iP_s)^g\le P_iP_s$ for some $g\in G$. Then $P_s$ and $P_s^g$ are Sylow $p_s$-subgroups of $P_iP_s$, so for some $k\in P_iP_s$, $P_s^{gk}=P_s$. Now $Q_i^{gk}P_s=(Q_iP_s)^{gk}\le P_iP_s\le H$. But $Q_i$ normalizes $P_s$, so $Q_i^{gk}$ normalizes $P_s^{gk}=P_s$. Therefore $Q_i^{gk}$ is a non-identity $p_i$-subgroup of $N_H(P_s)=P_s$, an impossibility.

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  • $\begingroup$ I can look up Hall's extension of Sylow's theorem, but I wouldn't even have known it existed if not for this answer. Can you recommend any good group-theory texts for someone like me who's never heard of this result? $\endgroup$
    – LSpice
    Dec 5, 2022 at 19:08
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    $\begingroup$ Most texts containing substantial amounts of finite group theory contain Hall's theorems. They should not require an elaborate run-up. I'd mention "The Theory of Groups" by Marshall Hall (and dedicated to Philip Hall), and "An Introduction to the Theory of Groups" by Rotman. Also "Finite Soluble Groups" by Doerk and Hawkes, "Finite Group Theory" by Gorenstein, "Finite Group Theory" by Aschbacher, "Finite Group Theory" by Isaacs, "The Theory of Finite Groups" by Kurzweil and Stellmacher. There are also massive texts by Huppert ("Endliche Gruppen I") and Suzuki "Group Theory I,II". $\endgroup$ Dec 5, 2022 at 20:06
  • $\begingroup$ Thank you! Though I knew many of these, I had found the ones that I tried (Aschbacher's and Huppert's, if I remember correctly—it was long ago) rather forbidding reading. Are you in a position to recommend any of them as particularly appropriate for self-study (for someone whose research is in algebraic groups, but hasn't previously worried overmuch about the structure of finite groups)? $\endgroup$
    – LSpice
    Dec 5, 2022 at 20:58
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    $\begingroup$ Try Rotman or Isaacs. $\endgroup$ Dec 6, 2022 at 2:16
  • $\begingroup$ Thank you very much! $\endgroup$ Dec 6, 2022 at 4:11

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