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An Extension of a Theorem of Janko on Finite Groups with Nilpotent Maximal Subgroups

Published online by Cambridge University Press:  20 November 2018

John W. Randolph
John W. Randolph*
Affiliation:
West Virginia University, Morgantown, West Virginia
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Throughout this paper G will denote a finite group containing a nilpotent maximal subgroup S and P will denote the Sylow 2-subgroup of S. The largest subgroup of S normal in G will be designated by core (S) and the largest solvable normal subgroup of G by rad(G). All other notation is standard.

Thompson [6] has shown that if P = 1 then G is solvable. Janko [3] then observed that G is solvable if P is abelian, a condition subsequently weakened by him [4] to the assumption that the class of P is ≦ 2 . Our purpose is to demonstrate the sufficiency of a still weaker assumption about P.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 550 - 552
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Alperin, J., Sylow intersections and fusion, J. Algebra 6 (1967), 222–241.Google Scholar
2. Feit, W., Characters of finite groups (Benjamin, New York, 1967).Google Scholar
3. Janko, Z., Verallgemeinerung eines satzes von B. Huppert una J. G. Thompson, Arch. Math. 12 (1961), 280–281.Google Scholar
4. Janko, Z., Finite groups with a nilpotent maximal subgroup, J. Austral. Math. Soc. 4 (1964), 449–451.Google Scholar
5. Randolph, J., Finite groups with solvable maximal subgroups, Proc. Amer. Math. Soc. 23 (1969), 490–492.Google Scholar
6. Thompson, J., Normal p-complements for finite groups, Math. Z. 72 (1960), 332–354.Google Scholar