## Abstract

We study the socle and the radical of a Mackey functor M for a finite group G over a field K (usually, of characteristic p > 0). For a subgroup H of G, we construct bijections between some classes of the simple subfunctors of M and some classes of the simple K over(N, -)_{G} (H)-submodules of M (H). We relate the multiplicity of a simple Mackey functor S_{H, V}^{G} in the socle of M to the multiplicity of V in the socle of a certain K over(N, -)_{G} (H)-submodule of M (H). We also obtain similar results for the maximal subfunctors of M. We then apply these general results to some special Mackey functors for G, including the functors obtained by inducing or restricting a simple Mackey functor, Mackey functors for a p-group, the fixed point functor, and the Burnside functor B_{K}^{G}. For instance, we find the first four top factors of the radical series of B_{K}^{G} for a p-group G, and assuming further that G is an abelian p-group we find the radical series of B_{K}^{G}.

Original language | English |
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Pages (from-to) | 2970-3025 |

Number of pages | 56 |

Journal | Journal of Algebra |

Volume | 321 |

Issue number | 10 |

DOIs | |

Publication status | Published - 15 May 2009 |

Externally published | Yes |

## Keywords

- Brauer quotient
- Burnside functor
- Loewy length
- Mackey algebra
- Mackey functor
- Maximal subfunctor
- Primordial subgroup
- Radical
- Restriction kernel
- Simple subfunctor
- Socle