### Abstract

Original language | English |
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Pages (from-to) | 35-52 |

Number of pages | 18 |

Journal | Journal of Algebra |

Volume | 535 |

Early online date | 26 Jun 2019 |

DOIs | |

Publication status | E-pub ahead of print - 26 Jun 2019 |

### Cite this

*Journal of Algebra*,

*535*, 35-52. https://doi.org/10.1016/j.jalgebra.2019.05.047

**Groups satisfying a strong complement property.** / Detomi, Eloisa ; Lucchini, Andrea; Moscatiello, Mariapia ; Spiga, Pablo; Traustason, Gunnar.

Research output: Contribution to journal › Article

*Journal of Algebra*, vol. 535, pp. 35-52. https://doi.org/10.1016/j.jalgebra.2019.05.047

}

TY - JOUR

T1 - Groups satisfying a strong complement property

AU - Detomi, Eloisa

AU - Lucchini, Andrea

AU - Moscatiello, Mariapia

AU - Spiga, Pablo

AU - Traustason, Gunnar

PY - 2019/6/26

Y1 - 2019/6/26

N2 - Let G=NH be a finite group where N is normal in G and H is a complement of N in G. For a given generating sequence (h1,…,hd) of H we say that (N,(h1,…,hd)) satisfies the strong complement property, if 〈h1 x1,…,hd xd〉 is a complement of N in G for all x1,…,xd∈N. When d is the minimal number of elements needed to generate H, and (N,(h1,…,hd〉)) satisfies the strong complement property for every generating sequence (h1,…,hd) with length d, then we say that (N,H) satisfies the strong complement property. In the case when |N| and |H| are coprime, we show that (N,H) can only satisfy the strong complement property if H is cyclic or if H acts trivially on N. We give on the other hand a number of examples that show this does not need to be the case when considering the strong complement property of (N,(h1,…,hd)) for a given fixed generating sequence. In the case when N and H are not of coprime order, we give examples where (N,H) satisfies the strong complement property and where H is not cyclic and does not act trivially on N.

AB - Let G=NH be a finite group where N is normal in G and H is a complement of N in G. For a given generating sequence (h1,…,hd) of H we say that (N,(h1,…,hd)) satisfies the strong complement property, if 〈h1 x1,…,hd xd〉 is a complement of N in G for all x1,…,xd∈N. When d is the minimal number of elements needed to generate H, and (N,(h1,…,hd〉)) satisfies the strong complement property for every generating sequence (h1,…,hd) with length d, then we say that (N,H) satisfies the strong complement property. In the case when |N| and |H| are coprime, we show that (N,H) can only satisfy the strong complement property if H is cyclic or if H acts trivially on N. We give on the other hand a number of examples that show this does not need to be the case when considering the strong complement property of (N,(h1,…,hd)) for a given fixed generating sequence. In the case when N and H are not of coprime order, we give examples where (N,H) satisfies the strong complement property and where H is not cyclic and does not act trivially on N.

U2 - 10.1016/j.jalgebra.2019.05.047

DO - 10.1016/j.jalgebra.2019.05.047

M3 - Article

VL - 535

SP - 35

EP - 52

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -