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The following question is about finite groups.

Let $G$ be a finite group and let $H, K \leqslant G$. We say that $H$ permutes with $K$ if $HK = KH$ and in this case $HK \leqslant G$.

The Symbol $\pi (G)$ denotes the set of all primes that divides the order of $G$.

I need to find an example of a finite group $G$, preferably $G$ is solvable, such that the following properties hold:

(1) $G$ has a subnormal subgroup $K$.

(2) There exists a prime $p \in \pi (G)$ such that $K$ does not permute with any Sylow $p$-subgroup of $G$.

(3) $G$ has a set of nilpotent Hall subgroups $\{H_1, H_2, \dots, H_n\}$ such that $\pi (G) = \bigcup_{i = 1}^n \pi (H_i)$ and $(|H_i|,|H_j|) =1$ for $i \neq j$ and $i, j = 1, 2, \dots n$.

(4) $K$ permutes with $H_i$ for all $H_i \in \{H_1, H_2, \dots, H_n\}$.

Remark: Condition (2) implies that $|\pi (H_i)| \geqslant 2$ for at least one $i$.

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  • $\begingroup$ Could you say something about why you need to find such an example? At first sight this looks like a very arbitrary list of properties. Also how far have you got with your search? $\endgroup$
    – Derek Holt
    Oct 7, 2017 at 12:23
  • $\begingroup$ @DerekHolt Here is what I Know: $|\pi (G)|>2$, this is clear by the above remark. $\endgroup$
    – user28083
    Oct 7, 2017 at 13:52
  • $\begingroup$ @DerekHolt $K$ is not nilpotent. Suppose $K$ is nilpotent and let $\pi = \pi (H_i)$. Then $K = K_\pi K_{\pi^\prime}$ and by (4), $K_{\pi^\prime}H_i = K_{\pi^\prime} K_\pi H_i = KH_i$. As $K_{\pi^\prime}$ is subnormal Hall of $K_{\pi^\prime}H_i$, then $K_{\pi^\prime} \lhd K_{\pi^\prime}H_i$. So, $K = K_\pi K_{\pi^\prime}$ permutes with every all Sylow subgroups of $H_i$ which are Sylow subgroups of $G$ for all $i$. This contradicts (2). $\endgroup$
    – user28083
    Oct 7, 2017 at 13:53
  • $\begingroup$ @DerekHolt I used GAP and search the groups of the orders from 1 to 1000. Also, I have searched some subgroups which are divisible by 4 primes and some which are divisible by 5 primes. But I did not have any luck with those. $\endgroup$
    – user28083
    Oct 7, 2017 at 13:53
  • $\begingroup$ @DerekHolt This is a research by that works some problems involving permutability that deal with the set mentioned in (3). degruyter.com/view/j/jgth.2015.18.issue-2/jgth-2014-0045/… $\endgroup$
    – user28083
    Oct 7, 2017 at 13:54

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