In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was introduced in Dwyer & Wilkerson (1994), making precise earlier notions of a mod p finite loop space.
A p-compact group is a pointed space BG, which is local with respect to mod p homology, and such the pointed loop space G = ΩBG has finite mod p homology.
One sometimes also refer to the p-compact group by G, but then one needs to keep in mind that the loop space structure is part of the data (which then allows one to recover BG).
For primes greater than 3, family 2 on the Shepard-Todd list will contain infinitely many exotic p-compact groups.
Using the classification, one can identify the compact Lie groups inside finite loop spaces, giving a homotopical characterisation of compact connected Lie groups: They are exactly those finite loop spaces that admit an integral maximal torus; this was the so-called maximal torus conjecture.