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CLASSIFICATION OF REFLECTION SUBGROUPS MINIMALLY CONTAINING
$p$-SYLOW SUBGROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 97 / Issue 1 / February 2018
- Published online by Cambridge University Press:
- 04 October 2017, pp. 57-68
- Print publication:
- February 2018
-
- Article
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- You have access
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Let a prime
$p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a 
$p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the 
$p$-Sylow subgroups are the groups of type 
$F_{4}$ when 
$p=2$ and 
$I_{2}(m)$ when 
$m\geq 6$ is even but not a power of 
$2$ for each odd prime divisor 
$p$ of 
$m$. The classification significantly reduces the cases required to describe the 
$p$-Sylow subgroups of finite real reflection groups.
