## Abstract

Let M be a Mackey functor for a finite group G. By the kernel of M we mean the largest normal subgroup N of G such that M can be inflated from a Mackey functor for G / N. We first study kernels of Mackey functors, and (relative) projectivity of inflated Mackey functors. For a normal subgroup N of G, denoting by P_{H, V}^{G} the projective cover of a simple Mackey functor for G of the form S_{H, V}^{G} we next try to answer the question: how are the Mackey functors P_{H / N, V}^{G / N} and P_{H, V}^{G} related? We then study imprimitive Mackey functors by which we mean Mackey functors for G induced from Mackey functors for proper subgroups of G. We obtain some results about imprimitive Mackey functors of the form P_{H, V}^{G}, including a Mackey functor version of Fong's theorem on induced modules of modular group algebras of p-solvable groups. Aiming to characterize subgroups H of G for which the module P_{H, V}^{G} (H) is the projective cover of the simple K over(N, -)_{G} (H)-module V where the coefficient ring K is a field, we finally study evaluations of Mackey functors.

Original language | English |
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Pages (from-to) | 1993-2029 |

Number of pages | 37 |

Journal | Journal of Algebra |

Volume | 319 |

Issue number | 5 |

DOIs | |

Publication status | Published - 1 Mar 2008 |

Externally published | Yes |

## Keywords

- Evaluation
- Faithful Mackey functor
- Fong's theorem
- Imprimitive Mackey functor
- Induction
- Inflation
- Kernel
- Mackey algebra
- Mackey functor
- Projective Mackey functor