Dear Forum,
Thank you all for your help. I seem to find myself running in circles with this topic. Could someone help me understand why it is not sufficient for a sporadic group G that if: 1) G = <g_1,g_2,g_3>, with g_i an element of a rational conjugacy class. 2) g_1*g_2*g_3 = 1 3) The symmetrised structure constant (as defined in the response by Thomas) equals 1. 4) that simply that for a sporadic group G, finding a triple generator <g_1,g_2,g_3> = G with g_1*g_2*g_3 = 1, and the symmetrised structure constant (as being defined in the previous response of Thomas) being equal to 1 does not sufficiently determine rational rigidity (i.e that G is a Galois group over Q)? I have looked at the book by Lax and at Page 132 (Theorem 2.5.19) and isn't the above essentially what it is saying; that if one can realize a centreless group (true for sporadic groups) as a rigid tuple of rational conjugacy classes then the group G is Galois over Q? Best, John Sent from Outlook<http://aka.ms/weboutlook> ________________________________ From: Hulpke,Alexander <alexander.hul...@colostate.edu> Sent: Saturday, August 19, 2017 8:45:13 AM To: johnathon simons Cc: forum@mail.gap-system.org; s...@math.rwth-aachen.de Subject: Re: [GAP Forum] Generating a group from a triple of elements. Dear Forum, Just a very brief note on one remark: > 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. My understanding (for details see the Book on representation theory by Lux and Pahlings, and ultimately — as they refer to it — the book by Malle and Matzat) is that the rigidity criterion only realizes M11 over a number field and further work is needed to obtain a rational realization from this. Regards, Alexander Hulpke > > 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
