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15.25 (2002)

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A finite group $G$ is said to be rational if every irreducible character of $G$ takes only rational values. Are the Sylow 2-subgroups of the symmetric groups $S_{2^n}$ rational?

Contributor: Ya. G. Berkovich

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Yes, they are. A Sylow 2-subgroup $T_n$ of $S_{2^n}$ is the wreath product of the one for $S_{2^{n-1}}$ with the group $C$ of order 2. If $T_n$ is rational, then the wreath product of $T_n$ with $C$ is also rational by Corollary 70 in (D. Kletzing Structure and representations of Q-groups (Lecture Notes in Math., 1084), Springer, Berlin, 1984). The result follows by induction. (A. Mann, Letter of 1 October 2002.)

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