Timeline for Irreducible mod-p representation of a semidirect product with trivial p-core
Current License: CC BY-SA 3.0
16 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 18, 2011 at 22:08 | vote | accept | Maurizio Monge | ||
| Nov 2, 2011 at 9:16 | comment | added | Derek Holt | I would still be interested in an answer to the original question - i.e. is the proposition true, and is there a reference? | |
| Nov 1, 2011 at 16:08 | vote | accept | Maurizio Monge | ||
| Dec 18, 2011 at 22:08 | |||||
| Nov 1, 2011 at 16:08 | comment | added | Maurizio Monge | By the way, I was noticing that in Bartel's proof $C$ is not required to be cyclic. | |
| Oct 31, 2011 at 21:53 | comment | added | Derek Holt | Yes, because the inflation restriction sequence $H^1(G/H,V^H) \to H^1(G,V) \to H^1(H,V)$ is exact - that one is always exact. | |
| Oct 31, 2011 at 21:04 | comment | added | Maurizio Monge | One last question: is it possible to see with the same methods that $G$ in $V\rtimes{}G$ is the unique complement, up to conjugacy? | |
| Oct 31, 2011 at 16:17 | comment | added | Alex B. | Dear Derek, indeed, for the exactness of inflation-restriction, one needs $H^1(H,V)=0$ (and the term on the right was not quite right), but since it's the same argument as for $H^2$, I might as well immediately use the fact that inflation is then an isomorphism, as Steve pointed out. Editing now... | |
| Oct 31, 2011 at 16:16 | history | edited | Alex B. | CC BY-SA 3.0 |
Simplified the argument a little bit, cleaned up the post
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| Oct 31, 2011 at 15:51 | comment | added | Derek Holt | That's true, but it doesn't really answer my question! | |
| Oct 31, 2011 at 15:43 | comment | added | Steve D | Instead of "2" above, I meant any $n$. | |
| Oct 31, 2011 at 15:42 | comment | added | Steve D | I believe in this case, since $|H|$ and $|V|$ are coprime, there is always an isomorphism between $H^2(G/H,V^H)$ and $H^2(G,V)$. | |
| Oct 31, 2011 at 15:23 | comment | added | Derek Holt | Where are you getting the exactness of that inflation-restriction sequence from? It is not exact in general. For example, if $G$ is a Klein 4-group, $|H| = 2$ and $|V|=2$, then $|H^2(G/H,V^H)| = |H^2(H,V)|=2$ but $H^2(G,V)| = 8$. | |
| Oct 31, 2011 at 15:18 | comment | added | Maurizio Monge | In any case thank you very much for illustrating how group cohomology can be used to prove this. | |
| Oct 31, 2011 at 15:08 | comment | added | Maurizio Monge | well, thanks, my proof once $V$ is known to be $C$-free my proof was even shorter: the extension by $H$ split by Schur-Zassenhaus, let $\sigma$ be a lifting of generator of $C$, then $\sigma^{p^k}$ is in the socle of $V$, but replacing it with $\sigma{}v$ for a suitable $v$ we can assure $(\sigma{}v)^{p^k}=\sigma^{p^k}N(v)$ to be $0$ if $V$ is free. I expected these facts to be very standard, if you had to prove it yourself in your paper I will probably do the same. | |
| Oct 31, 2011 at 13:30 | history | edited | Alex B. | CC BY-SA 3.0 |
Gave the full proof here, instead of just a reference.
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| Oct 31, 2011 at 13:21 | history | answered | Alex B. | CC BY-SA 3.0 |