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 $p^{\prime }$-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 $p^{\prime }$-order.