Geodesic
On rationality and 2-reflexiveness of wreath products of finite groups
Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 395-402

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A finite group $G$ is said to be rational if each its irreducible character acquires only rational values, and it is said to be 2-reflexive if each its element can be represented as a product of at most two involutions. We find necessary and sufficient conditions for the wreath of two finite groups be rational and 2-reflexive. Namely, we show that the wreath $H\wr K$ of two finite groups $H$ and $K$ is a rational (respectively, 2-reflexive) group iff $H$ is a rational (respectively, 2-reflexive) group and $K$ is an elementary Abelian 2-group. As a corollary, we obtain a description of all classical linear groups over finite fields of odd characteristic with rational and 2-reflexive Sylow 2-subgroups.
Keywords: wreath product, Sylow group, 2-reflexive group, irreducible character, classical linear group, dihedral group.
Mots-clés : rational group 
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     author = {S. G. Kolesnikov},
     title = {On rationality and 2-reflexiveness of wreath products of finite groups},
     journal = {Matemati\v{c}eskie zametki},
     pages = {395--402},
     publisher = {mathdoc},
     volume = {80},
     number = {3},
     year = {2006},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2006_80_3_a8/}
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S. G. Kolesnikov. On rationality and 2-reflexiveness of wreath products of finite groups. Matematičeskie zametki, Tome 80 (2006) no. 3, pp. 395-402. http://geodesic.mathdoc.fr/item/MZM_2006_80_3_a8/