On Burnside’s lemma

Herzog, Marcel

doi:10.1090/S0002-9939-1971-0274589-1

Proceedings of the American Mathematical Society

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On Burnside’s lemma
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by Marcel Herzog PDF

Proc. Amer. Math. Soc. 28 (1971), 379-380 Request permission

Abstract:

Burnside’s lemma on characters of finite groups is generalized, leading to the following theorem: if $G$ is a simple group of order divisible by exactly 3 primes, and if one of the Sylow subgroups of $G$ is cyclic, then for each Sylow subgroup $P$ of $G$ we have ${C_G}(P) = Z(P)$.

References

Richard Brauer, On simple groups of order $5\cdot 3^{a}\cdot 2^{b}$, Bull. Amer. Math. Soc. 74 (1968), 900–903. MR 236255, DOI 10.1090/S0002-9904-1968-12073-7
Walter Feit, Characters of finite groups, W. A. Benjamin, Inc., New York-Amsterdam, 1967. MR 0219636
Marcel Herzog, On finite groups with cyclic Sylow subgroups for all odd primes, Israel J. Math. 6 (1968), 206–216. MR 235037, DOI 10.1007/BF02760253