## Abstract

In 2011, Guralnick and Tiep proved that if G was a Chevalley group with Borel subgroup B and V an irreducible G-module in cross characteristic with V^{B}=0, then the dimension of H^{1}(G,V) is determined by the structure of the permutation module on the cosets of B. We generalise this theorem to higher cohomology and an arbitrary finite group, so that if H≤G such that O_{r′}(H)=O^{r}(H) and V^{H}=0 for V a G-module in characteristic r then dimH^{1}(G,V) is determined by the structure of the permutation module on cosets of H, and H^{n}(G,V) by Ext_{G}^{n−1}(V^{⁎},M) for some kG-module M dependent on H. We also determine Ext_{G}^{n}(V,W) for all irreducible kG-modules V, W for G∈{PSL_{2}(q),PGL_{2}(q),SL_{2}(q)} in cross characteristic.

Original language | English |
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Pages (from-to) | 347-379 |

Number of pages | 33 |

Journal | Journal of Algebra |

Volume | 595 |

DOIs | |

Publication status | Published - 1 Apr 2022 |

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