Timeline for Subgroup rank of finite simple groups
Current License: CC BY-SA 4.0
10 events
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| May 7, 2021 at 17:58 | comment | added | Richard Lyons | @YCor: Your conjecture is true. Use the Guralnick-Lucchini theorem in Geoff's answer. Let $G$ be the $F_{r^a}$-points of a simple algebraic group $\bar G$ of Lie rank $n$, and let $P$ be a Sylow $p$-subgroup of $G$. If $p\ne r$, $P\le N_{\bar G}(T)$ for some maximal torus $T$ of $\bar G$, so the $p$-subgroup rank of $P$ is at most $n+\log_2|W|$, independent of $a$ and $p$. If $p=r$, then $|P|\le p^{Ca}$ where $C$ is the number of positive roots, so the $p$-subgroup rank is at most $Ca$. | |
| May 4, 2021 at 16:57 | comment | added | user203598 | Dear @Yves, thank you for the comments. We seem to have given the same proof for the implication "bounded subgroup rank implies bounded Lie-rank and bounded field-rank." Here, my "Lie-rank" is your "rank," and my "field-rank" is your "degree over the prime field." I chose to use slightly unorthodox terminology in my original post to avoid mixing up the various different invariants. Apologies for the confusion. Concerning your conjecture about the reverse statement: I wondered the same after the posts of Geoff and Derek. Interesting question! | |
| May 4, 2021 at 15:42 | comment | added | YCor | It might be that a family of bounded rank finite simple groups has bounded subgroup rank iff the associated fields have bounded degree over their prime subfield. | |
| May 4, 2021 at 15:42 | comment | added | YCor | Here's my answer to another question about the notion of bounded rank finite simple group. Clearly bounded subgroup rank implies bounded rank. The converse is not true since $\mathrm{PSL}_2(\mathbf{F}_{p^n})$ contains a copy of $(\mathbf{Z}/p\mathbf{Z})^n$, and hence, for varying $n$ has unbounded subgroup rank. However $\mathrm{PSL}_2(\mathbf{F}_p)$ has bounded subgroup rank when the prime $p$ varies, by this answer. | |
| May 4, 2021 at 12:49 | vote | accept | user203598 | ||
| May 4, 2021 at 11:00 | history | edited | Francesco Polizzi | CC BY-SA 4.0 |
added 34 characters in body
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| May 4, 2021 at 10:55 | history | edited | YCor |
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| May 4, 2021 at 10:25 | answer | added | Geoff Robinson | timeline score: 3 | |
| May 4, 2021 at 9:47 | review | First posts | |||
| May 4, 2021 at 9:57 | |||||
| May 4, 2021 at 9:42 | history | asked | user203598 | CC BY-SA 4.0 |