[GAP Forum] Generating a group from a triple of elements.

[johnathon simons]

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


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________________________________
From: Hulpke,Alexander <Alexander.Hulpke at colostate.edu>
Sent: Saturday, August 19, 2017 8:45:13 AM
To: johnathon simons
Cc: forum at mail.gap-system.org; sam at 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,
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> John
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>
> [https://cdn.sstatic.net/Sites/math/img/apple-touch-icon@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 ...
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