Timeline for When is there a unique perfect group of order $n$?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jan 18, 2020 at 9:10 | comment | added | Geoff Robinson | I think that works, but the argument I had in mind was the following: G/Op′(G) has a self centralizing normal Sylow p-subgroup of order p (eg by Hall-Higman centralizer Lemma), so is isomorphic o a subgroup of order divisible by $p$ of a Frobenius group of order p(p−1), so is itself either a Frobenius group or cyclic of order $p$. If cyclic of order $p$, then G has a factor group of order p, while if the Frobenius complement is non-trivial, it is a cyclic subgroup of order dividing p−1 which is a homomorphic image of G. | |
| Jan 16, 2020 at 11:25 | comment | added | Sean Eberhard | Details of second paragraph (please correct me if you had something simpler in mind): If $G$ is $p$-solvable and has order $n$ then some quotient $G/N$ has a normal subgroup of order $p$, and this extension is split by Schur--Zassenhaus. Your two cases are whether this split product is direct or not. | |
| Jan 15, 2020 at 1:07 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
minor clarification
|
| Jan 15, 2020 at 0:03 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
Expanded to cover $n/2$ case
|
| Jan 14, 2020 at 19:17 | vote | accept | Sean Eberhard | ||
| Jan 14, 2020 at 19:17 | comment | added | Sean Eberhard | Nice, thank you! | |
| Jan 14, 2020 at 18:16 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
typo
|
| Jan 14, 2020 at 18:04 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
Expanded answer and explanation
|
| Jan 14, 2020 at 17:59 | history | edited | Geoff Robinson | CC BY-SA 4.0 |
Expanded answer and explanation
|
| Jan 14, 2020 at 17:41 | history | edited | LSpice | CC BY-SA 4.0 |
Names of articles, and minor typo fixes
|
| Jan 14, 2020 at 17:32 | history | answered | Geoff Robinson | CC BY-SA 4.0 |