On two theorems of Thompson

Zhang, Guang Xiang

doi:10.1090/S0002-9939-1986-0861754-6

On two theorems of Thompson
HTML articles powered by AMS MathViewer

by Guang Xiang Zhang PDF

Proc. Amer. Math. Soc. 98 (1986), 579-582 Request permission

Abstract:

Let $G$ be a finite group. Theorem. Let $P \in {\operatorname {Syl} _p}(G)$ with ${\Omega _1}(P) \leq Z(P)$. If ${N_G}(Z(P))$ has a normal $p$-complement, then so does $G$. Corollary. Let $M$ be a nilpotent maximal subgroup of $G$ and $P \in {\operatorname {Syl} _2}(M)$ with ${\Omega _2}(P) \leq Z(P)$. Then $G$ is solvable. This extends Thompson’s solvability theorem [9]. We also give two other results generalizing Thompson’s theorem.

References

Kevin Brown, Finite groups with nilpotent maximal $A$-invariant subgroups, Arch. Math. (Basel) 22 (1971), 146–149. MR 294496, DOI 10.1007/BF01222553
Miao Shêng Ch’ên, A note on Glauberman theorem, Tamkang J. Math. 7 (1976), no. 1, 111–114. MR 417279
Zhong-mu Chen, An extension of Burnside’s theorem, Chinese Math. 5 (1964), 81–84. MR 162847
Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. MR 166261
Daniel Gorenstein, Finite groups, 2nd ed., Chelsea Publishing Co., New York, 1980. MR 569209
Zvonimir Janko, Finite groups with a nilpotent maximal subgroup, J. Austral. Math. Soc. 4 (1964), 449–451. MR 0173703, DOI 10.1017/S1446788700025271
Hans Kurzweil, Endliche Gruppen, Hochschultext [University Textbooks], Springer-Verlag, Berlin-New York, 1977 (German). Eine Einführung in die Theorie der endlichen Gruppen. MR 0486072, DOI 10.1007/978-3-642-95313-2
John G. Thompson, Normal $p$-complements for finite groups, J. Algebra 1 (1964), 43–46. MR 167521, DOI 10.1016/0021-8693(64)90006-7
John Thompson, Finite groups with fixed-point-free automorphisms of prime order, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 578–581. MR 104731, DOI 10.1073/pnas.45.4.578
Helmut Wielandt, Zum Satz von Sylow, Math. Z. 60 (1954), 407–408 (German). MR 64769, DOI 10.1007/BF01187386