I'm looking that in the Fusion System categories, the p-subgroups that are essential, are centric (by definition) and radical (by implication of the definition), but I want to know if there is an example about a p-subgroup that is radical and centric, but not essential. Anyone knows an example?
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Yes, take any prime $p$, and consider the fusion system for ${\rm GL}(4,p)$ ( ie objects the $p$-subgroups of ${\rm GL}(4,p)$ morphisms induced by conjugations). Take a maximal parabolic $P$ such that $P/U \cong {\rm GL}(3,p)$, where $U$ is the unipotent radical of $P$. Then $U$ is centric and radical but is not essential (as ${\rm GL}(3,p)$ has no strongly $p$-embedded subgroup).
answered Apr 27, 2016 at 23:35
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$\begingroup$ Is there any other smaller than this one? (Only curiosity) $\endgroup$– iam_agfCommented Apr 28, 2016 at 0:49
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1$\begingroup$ Well, I suppose the parabolic $P$ itself is a smaller example. In general, the smallest example, whatever it is, will be the fusion system of genuine group $G$ with $U = O_{p}(G)$ elementary Abelian and $C_{G}(U) = U$, such that $G/U$ has no strongly embedded subgroup. A candidate might be the case that $p = 2$, $U$ elementary of order $8$, and $G/U \cong {\rm GL}(3,2)$, but I have not checked. $\endgroup$ Commented Apr 28, 2016 at 7:41
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$\begingroup$ I meant "strongly $p$-embedded" in the last result. $\endgroup$ Commented Apr 28, 2016 at 9:51
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1$\begingroup$ Yes, I am the author of that paper. Some parts of it could be proved much more efficiently! $\endgroup$ Commented May 4, 2016 at 17:28
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1$\begingroup$ I gave some references in the comment above. About parabolic subgroups, the book of Carter may be the closest of that list. You need to read about groups with a $(B,N)$-pair. As I said, there are many choices. There are good surveys, for example, by C.W. Curtis. $\endgroup$ Commented Jul 5, 2016 at 6:50