Timeline for Has any attempt been made to classify finite groupoids?
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| May 20, 2020 at 17:02 | comment | added | Ronnie Brown | I ought to mention that Ch 11 of the book T&G mentioned in my last comment has accounts of some of the work of Higgins and his student John Taylor on the algebra of groups acting on groupoids, using notions such as fibrations and coverings of groupoids.. | |
| Mar 30, 2015 at 14:04 | comment | added | Ronnie Brown | @Qiaochu Yuan: Groupoids equipped with a $G$-action where $G$ is a group occur in the study of orbit spaces, where the groupoid is the fundamental groupoid $\pi_1(X)$. See Chapter 11 of Topology and Groupoids, where the "orbit groupoid" is related to the fundamental groupoid of the orbit space! | |
| Mar 29, 2015 at 17:50 | history | edited | Qiaochu Yuan | CC BY-SA 3.0 |
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| Mar 29, 2015 at 17:44 | comment | added | Qiaochu Yuan | @David: hmm, you're right. My bad. | |
| Mar 29, 2015 at 16:33 | comment | added | John Shareshian | One can prove by elementary methods that if $N$ is a simple normal subgroup of the finite group $G$ with $G/N$ simple and some prime $p$ divides $|G/N|$ but does not divide $|Aut(N)|$, then $G$ is isomorphic with $N \times G/N$. In the case at hand, $p=13$ works. | |
| Mar 29, 2015 at 16:07 | comment | added | David Treumann | Extensions of $G$ by a nonabelian $M$ are classified by cohomology of $G$ with coefficients in what the ancients called a crossed module; i.e. by homotopy classes of maps from $BG$ to the delooping of $\mathrm{Aut}(BM)$. Since $M_{12}$ has no center, $\mathrm{Aut}(BM_{12}) \cong \mathrm{Out}(BM_{12}) \cong \mathbf{Z}/2$. That means there can be no nontrivial extensions of a group like $\mathrm{SL}_3(\mathbf{F}_3)$ or $\mathrm{PGL}_3(\mathbf{F}_3)$ by $M_{12}$, right? | |
| Mar 27, 2015 at 1:46 | history | answered | Qiaochu Yuan | CC BY-SA 3.0 |