Timeline for Generalisation of abelianisation using representation theory?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Dec 18, 2023 at 18:08 | history | edited | Sam Hopkins | CC BY-SA 4.0 |
added 60 characters in body
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| Dec 18, 2023 at 17:43 | history | became hot network question | |||
| Dec 18, 2023 at 17:36 | answer | added | Geoff Robinson | timeline score: 10 | |
| Dec 18, 2023 at 10:21 | comment | added | HenrikRüping | I just wanted to stress what YCor has already pointed out: $G=G_,m$ just means residually embeddable in $GL_m(k)$ and not embeddable in $GL_m(k)$. I am not sure in which direction the implication in the last paragraph goes, but i cannot be an if and only if. | |
| Dec 18, 2023 at 10:16 | answer | added | HenrikRüping | timeline score: 0 | |
| Dec 18, 2023 at 9:56 | comment | added | YCor | I'd be curious of a general description of those finite groups $G$ such that $G=G_2$, i.e., residually embeddable in $\mathrm{GL}_2(\mathbf{C})$. It consists of groups whose only nonabelian Jordan-Hölder factors are isomorphic to $A_5$, but not conversely, since $A_5$ itself is not in the class (unlike the binary icosahedral group of order 120). | |
| Dec 18, 2023 at 9:45 | comment | added | YCor | "I've read that if $G$ has a generating set of size $m$ then $G_m=G$" shouldn't be taken too seriously. Indeed, every finite simple group has a generating pair, while $G_2\neq G$ for most finite simple groups (furthermore for every $n$, every large enough nonabelian finite simple group has no nontrivial representation of dimension $\le n$). | |
| Dec 18, 2023 at 9:14 | history | asked | gimothytowers | CC BY-SA 4.0 |