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Some conditions which almost characterize Frobenius groups. (English) Zbl 0654.20019

Let \(G\) be a finite group with proper normal subgroup \(H\), and let \(\{\chi_ 1,\dots,\chi_ n\}\) be the irreducible characters of \(G\) not containing \(H\) in their kernel. Theorem 1: The following conditions are equivalent: (i) For each \(g\in G\setminus H\), \(gH\) is contained in a conjugacy class of \(G\); (ii) Each \(\chi_ i\) vanishes on \(G\setminus H\) and there exist positive integers \(\alpha_ 1,\dots,\alpha_ n\) such that \(\sum\alpha_ i\chi_ i\) is constant on \(H^{\#}\). Theorem 2: Suppose that the pair \((G,H)\) satisfies one (hence both) of the conditions in Theorem 1. Then one of the following holds: (a) \(G\) is a Frobenius group with kernel \(H\); (b) \(H\) is a \(p\)-group for some prime \(p\); or (c) \(G/H\) is a \(p\)-group for some prime \(p\).

MSC:

20D25 Special subgroups (Frattini, Fitting, etc.)
20D40 Products of subgroups of abstract finite groups
20C15 Ordinary representations and characters
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, \(\pi\)-length, ranks
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References:

[1] B. Huppert,Endliche Gruppen I, Springer-Verlag, Berlin-Heidelberg-New York, 1967. · Zbl 0217.07201
[2] W. B. Stewart,Largely fixed-point-free groups, to appear.
[3] M. Suzuki,On a class of doubly transitive groups, Ann. of Math. (2)75 (1962), 105–145. · Zbl 0106.24702
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