Alternating and symmetric groups


Theorem. [AFGV1, F]   A conjugacy class in a symmetric or alternating group $\mathbb S_m$ or $\mathbb A_m$, $m \ge 5$, that does not appear in the Table below. is not kthulhu, hence it collapses.

 Here $\mathcal F = \left\{p\in \mathbb N: p \text{ prime, } p = \frac{r^k-1}{r-1}, \text{ where } r \text{ is a prime power and } k\in \mathbb N\right\}$.

$\mathbf G$type Reference 
$\mathbb S_m$$(1^{m-2},2)$kthulhu, [AFGV1, Remark 4.2]
$\mathbb A_m$$(1^{m-3},3)$austere, idem
$\mathbb A_p$, $p =5,7$ or $\notin \mathcal F$ $(p)$sober, [F, Remark 3.2 (b)]
$\mathbb A_{p+1}$, $p =5$ or $\notin \mathcal F$ $(1,p)$sober, [F, Remark 3.2 (c)]
$\mathbb A_8$$(2^4)$austere, [AFGV1, Remark 4.2]
$\mathbb A_7$$(2^2,3)$idem
$\mathbb S_6$ $(2^3)$kthulhu, isomorphic to the class of type $(1^{4},2)$
$\mathbb A_6$ $(3^2)$, $(1^2,2^2)$austere, [AFGV1, Remark 4.2]
$\mathbb S_5$$(2,3)$sober, idem
$\mathbb A_5$ $(1,2^2)$idem






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