Timeline for Characters of finite groups with Sylow $p$-subgroups of order $p$
Current License: CC BY-SA 4.0
4 events
| when toggle format | what | by | license | comment | |
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| Jan 9, 2022 at 21:18 | comment | added | Chris H | Hi Nick, I don’t know of any proof actually, elementary or not, but my background isn’t very strong. It also seems that rational classes of $G$ having character values equal to the character degrees of the centraliser (up to sign) occur quite frequently, not just in this cyclic case, but I don’t know of any precise statements along these lines. | |
| Jan 9, 2022 at 14:40 | comment | added | Nick | Hi Chris, I agree with your statement. It is easy to deduce that the character values on $p$-elements are rational, but is there an "elementary" way to argue that the values can only be 0, 1 or -1 for irreducible characters as you stated? | |
| Jan 9, 2022 at 11:10 | comment | added | Chris H | This would seemingly follow from a proof of the observation that in your setup, the character values of irreps of $G$ on the conjugacy class of a nonzero element of $P$ are all $0$, $1$ or $-1$. I would be curious to see if this has an elementary proof too. | |
| Jan 9, 2022 at 2:50 | history | asked | Nick | CC BY-SA 4.0 |