Each of the pairs of non-rational conjugacy classes — the two containing elements of order 11 and the two containing elements of order 8 form a rational class (at least in the sense used by the GAP command) when combined. Hence the total of 8 (6 single conjugacy classes and 2 pairs). See the GAP documentation under RationalClass and RationalClasses for details.
> On 19 Aug 2017, at 16:06, johnathon simons <johnathonasim...@outlook.com> > wrote: > > Dear Thomas, > > > Thank you for your helpful comments on calculating the symmetrised structure > constant. > > > Perhaps my understanding of rational classes is incorrect, but I have taken > it from this equivalence: > https://math.stackexchange.com/questions/218302/a-conjugacy-class-c-is-rational-iff-cn-in-c-whenever-c-in-c-and-n-is-co > >> From my understanding of that, we would say a conjugacy class is rational if >> every entry of it in the character table is rational under the respective >> column. However, using my reasoning, if we look at M11, under the columns 8a >> and 8b it has entries of A = -sqrt(-2) and also underneath 11a and 11b it >> has entries B = (-1 - sqrt(-11)/2) both which are not rational and so that >> would mean M11 has only 6 rational classes (it has a total of 10 conjugacy >> classes) > > > But, using the "RationalClasses" function on GAP we know that M11 has 8 > rational classes. I'm certain I am mistaken in my reasoning/interpretation > of the character table and would very much be appreciative of an explanation. > > Essentially, all I am trying to do is find a triple of conjugacy classes > (that are rational) such that a triple (g_1, g_2, g_3) of elements satisfies > the rigidity condition of Thompson to realize the group M11 as Galois over Q. > > I am very much appreciative for all your help, > > > John > > > [https://cdn.sstatic.net/Sites/math/img/apple-touch-i...@2.png?v=4ec1df2e49b1]<https://math.stackexchange.com/questions/218302/a-conjugacy-class-c-is-rational-iff-cn-in-c-whenever-c-in-c-and-n-is-co> > > A conjugacy class $C$ is rational iff $c^n\\in C$ whenever > ...<https://math.stackexchange.com/questions/218302/a-conjugacy-class-c-is-rational-iff-cn-in-c-whenever-c-in-c-and-n-is-co> > math.stackexchange.com > Let $C$ be a conjugacy class of the finite group $G$. Say that $C$ is > rational if for each character $\chi: G \rightarrow \mathbb C$ of $G$, for > each $c\in C$, we ... > > > > > Sent from Outlook<http://aka.ms/weboutlook> > _______________________________________________ > Forum mailing list > Forum@mail.gap-system.org > http://mail.gap-system.org/mailman/listinfo/forum _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
