Geodesic
Rationality and strong reality for Sylow 2-subgroups of Weyl and alternating groups
Algebra i logika, Tome 44 (2005) no. 1, pp. 44-53.

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It is proved that values of all complex characters for a Sylow 2-subgroup $P$ of any Weyl group are rational, and every element of $P$ is a product of two involutions in $P$. Similar results hold also for Sylow 2-subgroups of alternating groups.
Keywords: Weyl group, alternating group, Sylow 2-subgroup. 
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S. G. Kolesnikov. Rationality and strong reality for Sylow 2-subgroups of Weyl and alternating groups. Algebra i logika, Tome 44 (2005) no. 1, pp. 44-53. http://geodesic.mathdoc.fr/item/AL_2005_44_1_a2/

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