Timeline for Nonisomorphic finite groups with isomorphic Sylow subgroups
Current License: CC BY-SA 4.0
15 events
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| May 9, 2023 at 13:12 | history | edited | semisimpleton | CC BY-SA 4.0 |
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| S May 9, 2023 at 13:10 | vote | accept | semisimpleton | ||
| May 9, 2023 at 13:10 | vote | accept | semisimpleton | ||
| S May 9, 2023 at 13:10 | |||||
| May 9, 2023 at 8:48 | comment | added | Geoff Robinson | (cont.) Once we know this, it follows that whenever $G$ is a finite group with all its (non-trivial) Sylow subgroups self normalizing, then $G$ has a normal subgroup of prime index (clear if $|G|$ has at most two prime divisors, and if more than two prime divisors, then $G$ has a self-normalizing Sylow $q$-subgroup for some prime $q \geq 5$, and $G$ has a normal subgroup $H$ of index $q$. But now $H$ ( and also $G$) must be a $q$-group, otherwise we take a orime $r \neq q$ dividing $|H|$, and we have $G = HN_{G}(R)$ for $R \in {\rm Syl}_{r}(H)$. Then $G = HR = H$, contradiction. | |
| May 9, 2023 at 8:38 | comment | added | Geoff Robinson | This is a deep result, and unless you can locate the proceedings, you may have trouble finding it. It relies on a very deep theorem of Glauberman (12.5 in the above proc.), that if $p>3$ is a prime, and $G$ is a finite group with $N_{G}(S)/C_{G}(S)$ a $p$-group for $S ( \neq 1) \in {\rm Syl}_{p}(G)$, then $G$ has a factor group of order $p$. This uses the $K_{\infty}$-subgroups of Glauberman ..... | |
| May 9, 2023 at 4:41 | comment | added | semisimpleton | I didn't know about that theorem of Glauberman's! Thanks, I'll check it out. | |
| May 8, 2023 at 13:23 | comment | added | Geoff Robinson | I'm sure you know this, but the theorem that finite solvable groups have a unique conjugacy class of nilpotent self-normalizing subgroups is due to R. Carter. Also, G. Glauberman proved that if $G$ is a finite group whose Sylow subgroups are all self-normalizing is a $q$-group for some prime $q$. The latter is Corollary 12.6 in Glauberman's article in the Conference Proceedings " Finite Simple Groups", eds M.B. Powell and G. Higman, Academic Press, 1971. | |
| May 8, 2023 at 10:51 | history | became hot network question | |||
| May 8, 2023 at 9:36 | history | edited | YCor |
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| May 8, 2023 at 8:49 | answer | added | Sean Eberhard | timeline score: 9 | |
| May 8, 2023 at 7:55 | answer | added | Dave Benson | timeline score: 12 | |
| May 8, 2023 at 4:15 | history | edited | semisimpleton | CC BY-SA 4.0 |
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| May 8, 2023 at 4:12 | comment | added | semisimpleton | No, but I've edited my post to include some thoughts about finding such a pair. | |
| May 8, 2023 at 3:06 | comment | added | LSpice | Do you know a pair of non-isomorphic, locally isomorphic groups? | |
| May 8, 2023 at 2:49 | history | asked | semisimpleton | CC BY-SA 4.0 |