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An analogue to Glauberman's $ ZJ$-theorem

Author: Bernd Stellmacher
Journal: Proc. Amer. Math. Soc. 109 (1990), 925-929
MSC: Primary 20D20; Secondary 20D25
Erratum: Proc. Amer. Math. Soc. 114 (1992), null.
MathSciNet review: 1013982
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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 [Enhancements On Off] (What's this?)

  • [1] G. Glauberman, A characteristic subgroup of a $ p$-stable group, Canad. J. Math. 20 (1968), 1101-1135. MR 0230807 (37:6365)
  • [2] D. Go1dschmidt, Automorphisms of trivalent graphs, Ann. of Math. 111 (1980), 377-406. MR 569075 (82a:05052)
  • [3] D. Gorenstein, Finite groups, Harper & Row, New York. MR 0231903 (38:229)
  • [4] A. G. Kurosh, Theory of groups, Chelsea, New York, 1955.

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