An analogue to Glauberman’s 𝑍𝐽-theorem

Stellmacher, Bernd

doi:10.1090/S0002-9939-1990-1013982-8

An analogue to Glauberman’s $ZJ$-theorem
HTML articles powered by AMS MathViewer

by Bernd Stellmacher PDF

Proc. Amer. Math. Soc. 109 (1990), 925-929 Request permission

Erratum: Proc. Amer. Math. Soc. 114 (1992), 588.

Abstract:

Let $P$ be a finite $p$-group, $p$ an odd prime. Using certain versions of $p$-stability it is shown that there exists a nontrivial characteristic subgroup $W$ in $P$ that is normal in every finite $p$-stable group $G$ satisfying ${C_G}({O_p}(G)) \leq {O_p}(G)$ and $P \in {\text {Sy}}{{\text {l}}_p}(G)$. Moreover, $W$ contains every abelian subgroup of $P$ normalized by $W$.

References

George Glauberman, A characteristic subgroup of a $p$-stable group, Canadian J. Math. 20 (1968), 1101–1135. MR 230807, DOI 10.4153/CJM-1968-107-2
David M. Goldschmidt, Automorphisms of trivalent graphs, Ann. of Math. (2) 111 (1980), no. 2, 377–406. MR 569075, DOI 10.2307/1971203
Daniel Gorenstein, Finite groups, Harper & Row, Publishers, New York-London, 1968. MR 0231903

Theory of groups