Timeline for Embedding $S_3$ into $Aut(F_2)$
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| May 23, 2011 at 2:42 | comment | added | DavidLHarden | Here is an outline of a proof that if $G$ is a $p$-group, then $|G| \equiv c(G) \mod{(p-1)(p^{2}-1)}$: As noted before, this is already proven if $p=2$, so assume $p$ is odd. $(x,y)$ is a non-commuting pair iff $(y,x)$ is, and iff $(x^{k},y)$, where $k$ denotes a nonmultiple of $p$, is. These imply that $|G|-c(G)$ is a multiple of $2(p-1)^{2}$. Character-theoretically, $|G|$ is a sum of $c(G)$ powers of $p^{2}$. Reduce modulo $p^{2}-1$ and use complex conjugate pairing to conclude that $|G|-c(G)$ is a multiple of $2(p^{2}-1)$. These imply that $|G|-c(G)$ is a multiple of $(p-1)(p^{2}-1)$. | |
| May 16, 2011 at 22:35 | history | undeleted | DavidLHarden | ||
| May 16, 2011 at 22:35 | history | edited | DavidLHarden | CC BY-SA 3.0 |
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| May 16, 2011 at 22:00 | history | deleted | DavidLHarden | ||
| May 16, 2011 at 22:00 | history | answered | DavidLHarden | CC BY-SA 3.0 |