### Abstract

Glauberman's Z*-theorem and analogous statements for odd primes show that, for any prime p and any finite group G with Sylow p-subgroup S , the centre of G/Op′(G)G/Op′(G) is determined by the fusion system FS(G)FS(G). Building on these results we show a statement that seems a priori more general: For any normal subgroup H of G with Op′(H)=1Op′(H)=1, the centralizer CS(H)CS(H) is expressed in terms of the fusion system FS(G)FS(G) and its normal subsystem induced by H.

Original language | English |
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Pages (from-to) | 511-514 |

Number of pages | 4 |

Journal | Journal of Algebra |

Volume | 439 |

Early online date | 16 Jul 2015 |

DOIs | |

Publication status | Published - 1 Oct 2015 |

### Keywords

- finite groups
- fusion systems
- Glauberman's Z*-theorem

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## Cite this

Henke, E., & Semeraro, J. (2015). Centralizers of normal subgroups and the Z*-theorem.

*Journal of Algebra*,*439*, 511-514. https://doi.org/10.1016/j.jalgebra.2015.06.027