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The Artin exponent of finite groups. (English) Zbl 0931.20010
The following main result is proved: Let \(G\) be a finite \(p\)-group and \(e(G)\) the Artin exponent induced from the elementary Abelian subgroup of \(G\). One has the following: (1) If \(G\) is Abelian then \(e(G)=|G:U|\), where \(U\) is the maximal elementary Abelian subgroup of \(G\). (2) If \(G\) is a quaternion or dihedral group then \(e(G)=2\), and \(e(G)=4\) if \(G\) is the semidihedral group. (3) In all other cases \(e(G)=|G|/p\).
MSC:
20C15 Ordinary representations and characters
20D15 Finite nilpotent groups, \(p\)-groups
19A22 Frobenius induction, Burnside and representation rings
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References:
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