This paper provides an excellent summary of the recent ground-breaking work of W. G. Dwyer and C. W. Wilkerson [Ann. of Math. (2) 139 (1994), no. 2, 395–442; MR1274096 (95e:55019)] on p-compact groups. A p-compact group is an Fp-local space BX such that ΩBX is Fp-finite. In other words a p-compact group is a sort of “Fp-local finite loop space”, and as such a generalization of a compact Lie group. In this article, the author carefully explains the historical background of the problem, emphasizing throughout how the results obtained by Dwyer and Wilkerson for p-compact groups generalize known results for compact Lie groups. He also indicates how several of their most important results are proved. The author begins by providing examples of p-compact groups and defining p-compact group morphisms. He then discusses homotopy fixed point spaces, centralizers, algebraic Smith theory, and maximal tori and Weyl groups in the context of p-compact groups. He concludes with a survey of what is known so far about the classification of p-compact groups. Exercises are suggested throughout the paper, encouraging the ambitious reader to hone his understanding of the definitions and results presented
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