The classical representation theory of finite and compact groups is built around character theory and the classification of irreducible representations. A fundamental feature of this theory is complete reducibility: every representation decomposes into a direct sum of irreducible representations.

The representation theory of finite groups over fields of positive characteristic p, however, is far more subtle, since complete reducibility generally fails. In this setting, Sylow p-subgroups, together with homological and geometric methods such as support varieties, play a central role.

Supergroups and superalgebras arise naturally in the mathematical foundations of supersymmetry in theoretical physics, where ordinary commuting variables are combined with anticommuting variables (fermions). In this talk, we will explore how ideas from modular representation theory reappear in the study of representations of supergroups. We will discuss “super” analogues of the Sylow theorems and several related classical results in this new setting.