Timeline for Finite groups with number of generators strictly less than number of relations
Current License: CC BY-SA 4.0
9 events
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| Sep 30, 2022 at 16:56 | comment | added | HJRW | @YCor: your second comment seems like it would make a pretty good answer to me! | |
| Sep 30, 2022 at 6:57 | comment | added | YCor | And in general (with the |relators|$-$|generators| sign convention) the deficiency of a finite group is $\ge$ that of its Schur multiplier. See also this paper by Gardam (publ. Bull LMS 2017) for examples of finite groups with arbitrary positive deficiency. | |
| Sep 30, 2022 at 6:42 | comment | added | YCor | In general the "quest" is rather to find groups of deficiency zero among finite groups. See for instance this 1996 paper by Havas, Newman, and O'Brien. They mention in particular that a theorem of Golod-Shafarevich says that a finite $p$-group with deficiency zero is 3-generated (whether this holds for all finite groups is mentioned there as open). | |
| Sep 30, 2022 at 0:40 | comment | added | Benjamin Steinberg | The lingo for this kind of thing is deficiency. | |
| Sep 30, 2022 at 0:32 | comment | added | Will Sawin | $(\mathbb Z/p)^n$ is an example for all $n>1$, which can be proven using mod $p$ cohomology. | |
| Sep 29, 2022 at 23:56 | comment | added | Yiftach Barnea | You should read about the Golod-Shafarevich inequality. | |
| Sep 29, 2022 at 22:45 | comment | added | LSpice |
TeX note: please use $\langle\rangle$ \langle\rangle instead of $<>$ <>. It is also often preferable (though, unlike \langle\rangle, probably not universally agreed) to use \mid for a divider rather than |. (The results are $\langle a \mid a^n\rangle$ vs. $<a|a^n>$.) I have edited accordingly.
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| Sep 29, 2022 at 22:43 | history | edited | LSpice | CC BY-SA 4.0 |
TeX; deleted "thanks"
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| Sep 29, 2022 at 22:37 | history | asked | gola vat | CC BY-SA 4.0 |