The $D^{\Delta}$-pair biset category

Abstract

Let $p$ be a prime number. A $D^{\Delta}$-pair is a pair consisting of a finite $p$-group and a $p'$-automorphism of the group. In this paper, we introduce diagonal $D^\Delta$-pair bisets and a category whose objects are $D^\Delta$-pairs and whose morphism groups are Grothendieck groups of diagonal $D^\Delta$-pair bisets. Our main result shows that, over suitable coefficient rings, the category of diagonal $p$-permutation functors is equivalent to the category of linear functors on a natural quotient of this new category. In this way, diagonal $p$-permutation functors can be studied through a category built only from finite $p$-groups and their automorphisms of $p'$-order.