Can anyone please tell me The relationship between $p$-solvable Group and solvable group.and find an example of a $p$-solvable group that is not solvable group or vice-versa.
1 Answer
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A finite group is solvable if an only if it is $p$-solvable for every prime $p$. An example of a $p$-solvable group which is not solvable is the semidirect product $G = VSL(2,5)$, where $V$ is an elementary Abelian group of order $121$, and $SL(2,5)$ acts faithfully and irreducibly as a group of linear transformations on $V$. The group $G$ is $11$-solvable, but is not solvable.
Later edit: Easier, but perhaps less interesting, is that a non-Abelian finite simple group $G$ is $p$-solvable if and only if the prime $p$ does not divide $|G|$. More generally, any finite group $G$ is $p$-solvable for all but finitely many primes $p$.
(Recall that a finite group $G$ is $p$-solvable for a prime $p$ if and only if every composition factor of $G$ is either of order $p$ or of order prime to $p$. It is in fact true that a finite group $G$ is solvable if and only if $G$ is $2$-solvable, but this requires the very deep Feit-Thompson Theorem that groups of odd order are solvable).
answered Mar 18, 2016 at 12:12
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$\begingroup$ Nice answer, but I can't help but feel that the question would have been a better fit on MSE (though coming up with a non-trivial example of a non-solvable but $p$-solvable group is not trivial). $\endgroup$ Commented Mar 18, 2016 at 12:41
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$\begingroup$ This is probably a case where group theorists know what a p-solvable group is, but lots of non-specialists don't. I agree it's not research level for finite group-theorists! If people ask non-trivial questions in what seems to be good faith, I am usually not inclined to tell them to go elsewhere, but mostly leave my answers as comments. $\endgroup$ Commented Mar 18, 2016 at 12:46
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$\begingroup$ I agree that it is a somewhat specialized concept. But the first google hit (for me at least) is the groupprops page on the topic, which states the connection. But again, the example (and how non-trivial it can be to come up with) makes me not want to move the question. $\endgroup$ Commented Mar 18, 2016 at 12:49