Timeline for Roots of permutations
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2019 at 16:54 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
Added another remark
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| Mar 14, 2019 at 12:57 | comment | added | Alex B. | Our comments overlapped, we had been typing at the same time. I had also noticed that among the aforementioned small groups, in the vast majority of them (all but 23), the square root counting function attains its maximum at a central element. Your argument offers an explanation of this observation. | |
| Mar 14, 2019 at 12:33 | comment | added | Geoff Robinson | Not sure if you have now seen the new edit, which reduces the problem to the essential case of a central involution which has strictly more square roots than the identity. For groups like ${\rm SL}(2,q)$ ($q$ odd) I think it's obvious that the central involution has more square roots than the identity since the identity has only two square roots,whereas there are already at least six elements of order $4$ in the Sylow $2$-subgroup. But the character theoretic condition also shows this easily too. | |
| Mar 14, 2019 at 11:57 | comment | added | Alex B. | Are there any general results regarding Frobenius-Schur indicators of element centralisers in finite groups? In particular, are there any infinite families of groups that may have symplectic representations but for which your criterion kicks in? | |
| Mar 14, 2019 at 11:57 | comment | added | Alex B. | That's nice! A data point, in particular in connection with my related question (link in my answer): out of the 1911 groups of order less than 150 that have a symplectic representation, in 236 the square root counting function takes its maximum at the identity. Out of those, 8 satisfy your criterion, i.e. there is no $y$ and $\mu\in {\rm Irr}_{C_G(y)}$ with Frobenius-Schur indicator $-1$ and such that $y{\rm ker}\mu$ has order $2$ in $C_G(y)/{\rm ker}\mu$. All 8 are groups of order $128$. | |
| Mar 14, 2019 at 11:55 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
Added more remarks
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| Mar 13, 2019 at 20:59 | history | answered | Geoff Robinson | CC BY-SA 4.0 |