[GAP Forum] Generating a group from a triple of elements.
Stephen Linton
sl4 at st-andrews.ac.uk
Sat Aug 19 18:54:11 BST 2017
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 <johnathonasimons at outlook.com> wrote:
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> Dear Thomas,
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> Thank you for your helpful comments on calculating the symmetrised structure constant.
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> 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
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>> 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)
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> 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.
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> 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.
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> 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>
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> 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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