Extension groups between simple Mackey functors

Abstract

In order to better understand the structure of indecomposable projective Mackey functors, we study extension groups of degree 1 between simple Mackey functors. We explicitly determine these groups between simple functors indexed by distinct normal subgroups. We next study the conditions under which it is possible to restrict ourselves to that case, and we give methods for calculating extension groups between simple Mackey functors which are not indexed by normal subgroups. We then focus on the case where the simple Mackey functors are indexed by the same subgroup. In this case, the corresponding extension group can be embedded in an extension group between modules over a group algebra, and we describe the image of this embedding. In particular, we determine all extension groups between simple Mackey functors for a p-group and for a group that has a normal p-Sylow subgroup. Finally, we compute higher extension groups between simple Mackey functors for a group that has a p-Sylow subgroup of order p.