Timeline for Embeddings of finite groups into GL(n,Q_p)
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 24, 2013 at 18:39 | vote | accept | François Brunault | ||
| Apr 24, 2013 at 18:39 | comment | added | François Brunault | Thank you very much for taking the time to explain in more details! | |
| Apr 24, 2013 at 14:56 | comment | added | Socky | That's true, although one has to be a little careful by what one means by the reduction of a character mod $p$. If $\chi$ is faithful, there's a naive sense in which the reduction mod $p$ of $p \cdot \chi$ is trivial! So I re-wrote the argument in a slightly different way. | |
| Apr 24, 2013 at 14:52 | history | edited | Socky | CC BY-SA 3.0 |
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| Apr 23, 2013 at 12:31 | comment | added | François Brunault | Thanks for your answer! Your nice example shows that one can refine the question by looking only at those embeddings of $G$ into $\mathrm{GL}_n(\mathbf{Q}_p)$ which are compatible with a given complex faithful character of $G$. Regarding your argument, I don't see why the existence of a faithful $\mathbf{Q}_p$-valued character implies that $G$ embeds into $\mathrm{GL}_n(\mathbf{F}_p)$. Is it where you use your result about reps over finite fields? If so then why is the reduction mod $p$ still faithful? | |
| Apr 22, 2013 at 20:01 | history | answered | Socky | CC BY-SA 3.0 |