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An application of Burnside rings in elementary finite group theory. (English) Zbl 0752.19001
Let \(C\) be a finite cyclic group and \(G\) a finite group of the same order. The authors exhibit a canonical map (called the Frobenius-Wielandt homomorphism) from the Burnside ring \(\Omega(C)\) of \(C\) into the Burnside ring \(\Omega(G)\). Then many results from elementary finite group theory are simple consequences of the existence of this map [cf. B. Huppert, Endliche Gruppen. I (1967; Zbl 0217.07201), p. 34; and B. Wagner, Bayreuth. Math. Schr. 6, 23-32 (1980; Zbl 0452.20001)]. An extension to profinite groups is indicated.

MSC:
19A22 Frobenius induction, Burnside and representation rings
20D60 Arithmetic and combinatorial problems involving abstract finite groups
20E18 Limits, profinite groups
20C15 Ordinary representations and characters
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