Timeline for Finite groups with few conjugacy classes of maximal subgroups
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 10, 2018 at 19:41 | history | edited | Derek Holt | CC BY-SA 4.0 |
added 3172 characters in body
|
| Dec 10, 2018 at 15:03 | comment | added | Derek Holt | No, you are not being dense, I am using a result about maximal subgroups of subgroups of wreath products here. I will add an explanation to my answer, but that will have to wait until this evening. | |
| Dec 10, 2018 at 14:30 | comment | added | Nick Gill | Derek, great comments. Thanks. One thing though: it's not completely clear to me that whenever, say, $A_n$, is a section of a finite group $G$, then its intransitive maximals will yield distinct maximals for $G$... Am I just being dense? | |
| Dec 10, 2018 at 11:52 | comment | added | Derek Holt | I think that the answer to your second question is yes if you replace composition factor by chief factor. | |
| Dec 10, 2018 at 10:51 | comment | added | Derek Holt | The answer to your first question is yes, essentially because the number of parabolic maximal subgroups of groups of Lie type increases with their Lie rank. For example ${\rm PSL}_n(q)$ has $n-1$ of these. And $A_n$ has about $n/2$ intransitive maximal subgroups with two orbits. But $A_5 \wr C_p$ has $5$ classes of maximal subgroups for all primes $p$, so the answer to the second question is no. | |
| Dec 10, 2018 at 10:32 | comment | added | Nick Gill | Thanks Derek, this is really interesting. Let me idly speculate on the basis of 2 minutes of thought: do you think there could be a function $f(c)$ such that if $G$ is a group with at most $c$ conjugacy classes of maximal subgroups, then all simple groups involved with $G$ are of Lie type of rank at most $f(c)$, or alternating of size at most $f(c)$ (or sporadic)? I wonder if one could also bound the number of non-abelian compostion factors by a function of $c$? (I realise this is asymptotics which is not in the spirit of the original post...) | |
| Dec 10, 2018 at 8:41 | history | edited | Derek Holt | CC BY-SA 4.0 |
added 103 characters in body
|
| Dec 10, 2018 at 8:05 | history | edited | Derek Holt | CC BY-SA 4.0 |
added 540 characters in body
|
| Dec 9, 2018 at 22:03 | history | answered | Derek Holt | CC BY-SA 4.0 |