## Details

Original language | English |
---|---|

Pages (from-to) | 931-949 |

Number of pages | 19 |

Journal | Journal of group theory |

Volume | 26 |

Issue number | 5 |

Early online date | 11 May 2023 |

Publication status | Published - 1 Sept 2023 |

## Abstract

Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and (|N|, |G: H|) = 1. If N is abelian, a theorem of Gaschütz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z (N) ∩ N ′ ≠ 1. For metabelian groups N, the condition Z (N) ≠ N ′ = 1 implies the existence of complements. Finally, if N is perfect and centerless, then Gaschütz' theorem holds for N if and only if Inn (N) has a complement in Aut (N).

## ASJC Scopus subject areas

- Mathematics(all)
**Algebra and Number Theory**

## Cite this

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**On the converse of Gaschütz' complement theorem.**/ Sambale, Benjamin.

In: Journal of group theory, Vol. 26, No. 5, 01.09.2023, p. 931-949.

Research output: Contribution to journal › Article › Research › peer review

*Journal of group theory*, vol. 26, no. 5, pp. 931-949. https://doi.org/10.48550/arXiv.2303.00254, https://doi.org/10.1515/jgth-2022-0178

}

TY - JOUR

T1 - On the converse of Gaschütz' complement theorem

AU - Sambale, Benjamin

N1 - Funding statement: The work is supported by the German Research Foundation (SA 2864/1-2 and SA 2864/4-1).

PY - 2023/9/1

Y1 - 2023/9/1

N2 - Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and (|N|, |G: H|) = 1. If N is abelian, a theorem of Gaschütz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z (N) ∩ N ′ ≠ 1. For metabelian groups N, the condition Z (N) ≠ N ′ = 1 implies the existence of complements. Finally, if N is perfect and centerless, then Gaschütz' theorem holds for N if and only if Inn (N) has a complement in Aut (N).

AB - Let N be a normal subgroup of a finite group G. Let N ≤ H ≤ G such that N has a complement in H and (|N|, |G: H|) = 1. If N is abelian, a theorem of Gaschütz asserts that N has a complement in G as well. Brandis has asked whether the commutativity of N can be replaced by some weaker property. We prove that N has a complement in G whenever all Sylow subgroups of N are abelian. On the other hand, we construct counterexamples if Z (N) ∩ N ′ ≠ 1. For metabelian groups N, the condition Z (N) ≠ N ′ = 1 implies the existence of complements. Finally, if N is perfect and centerless, then Gaschütz' theorem holds for N if and only if Inn (N) has a complement in Aut (N).

UR - http://www.scopus.com/inward/record.url?scp=85163134466&partnerID=8YFLogxK

U2 - 10.48550/arXiv.2303.00254

DO - 10.48550/arXiv.2303.00254

M3 - Article

AN - SCOPUS:85163134466

VL - 26

SP - 931

EP - 949

JO - Journal of group theory

JF - Journal of group theory

SN - 1433-5883

IS - 5

ER -