Some simple biset functors

Abstract

Let 𝑝 be a prime number, let 𝐻 be a finite 𝑝-group, and let 𝔽 be a field of characteristic 0, considered as a trivial F ⁢ Out ⁢ ( H ) \mathbb{F}\mathrm{Out}(H) -module. The main result of this paper gives the dimension of the evaluation S H , F ⁢ ( G ) S_{H,\mathbb{F}}(G) of the simple biset functor S H , F S_{H,\mathbb{F}} at an arbitrary finite group 𝐺. A closely related result is proved in the last section: for each prime number 𝑝, a Green biset functor E p E_{p} is introduced, as a specific quotient of the Burnside functor, and it is shown that the evaluation E p ⁢ ( G ) E_{p}(G) is a free abelian group of rank equal to the number of conjugacy classes of 𝑝-elementary subgroups of 𝐺.