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Does there exist a quasisimple group $G$ and an odd prime $p$ such that $G$ has cyclic Sylow $p$-subgroups and a weakly real element of $p$-power order?

From Strongly real elements of odd order in sporadic finite simple groups the only sporadic finite simple group which has weakly real $2$-regular elements is McL. But McL does not have cyclic Sylow $p$-subgroups for the prime orders $3$ and $5$ of the weakly real elements.

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    $\begingroup$ What does "weakly real" mean? The cited post only defines real and strongly real. $\endgroup$ – Derek Holt Jun 13 '18 at 10:07
  • $\begingroup$ An element $g$ is weakly real in a group $G$ if it is conjugate to $g^{-1}$ in $G$, but $g^t\ne g^{-1}$ for any involution $t$ in $G$. So weakly real means real but not strongly real. $\endgroup$ – John Murray Jun 13 '18 at 10:13
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${\rm SL}(2,q)$ for odd $q$ has a unique element of order $2$, which is central. So for any odd prime $p$ dividing $q-1$ or $q+1$ it has a cyclic Sylow $p$-subgroup $P$ and a generator of $P$ is conjugate to its inverse but not by an involution. The smallest example is ${\rm SL}(2,5)$ with $p=3$, and $p=5$ also works in this case.

There are other examples. Normalizers of cyclic Sylow $p$-subgroups acting irreducibly in symplectic groups seem to be a source of examples. The results I have are just from computer calculations, but they should not be hard to prove more generally. I would guess that you might find similar examples in other classical groups.

Specific examples are ${\rm Sp}(4,q)$ for $q=3,5,7$ with $p=5,13,5$ respectively, ${\rm Sp}(6,q)$ with $p=7$, and ${\rm Sp}(8,3)$ with $p=41$.

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  • $\begingroup$ Thanks Derek, that was easy! Please let me know if you have other examples for other finite groups of Lie type. $\endgroup$ – John Murray Jun 13 '18 at 10:43
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Regarding the question in your comment on Derek's answer, a 1978 paper of Bob Griess ( Quarterly Journal) proves that the only quasisimple group ( other than an ${\rm SL}(2,q))$ with all involutions central is the double cover of ${\rm A}_{7}.$ A Sylow $5$-subgroup of this double cover contains a weakly regular element of order $5$. This (with Derek's answer) seems to exhaust the quasisimple examples in which all involutions are central. But of course there might ( a priori at least) be weakly regular $p$-elements in other quasisimple groups where the involutions are not all central.

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    $\begingroup$ Here is an infinite source of examples for the double covers of alternating groups. Let $p$ be an odd prime and let $g$ be a $p$-element of $2.A_p$ whose image in $A_p$ is the $p$-cycle $(1,..,p)$. In particular $\langle g\rangle$ is a Sylow $p$-subgroup of $2.A_p$. Each involution in $A_p$ which inverts $(1,..,p)$ is a product of $m:=(p-1)/2$ transpositions. Suppose that $4\not\div m$. Then there is no involution in $2.A_p$ whose image in $A_p$ is a product of $m$ transpositions. So $g$ is a weakly real $p$-element in $2.A_p$ if $p\not\equiv1$ (mod $8$). $\endgroup$ – John Murray Jun 13 '18 at 12:28

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