Cohomology of wreath products

Abstract Wreath products (defined below) seem to be ubiquitous in group theory and its applications. They appear implicitly in the work of 19th century group theorists, but were first defined explicitly by Polya when studying combinatorial problems associated with classifying organic molecules (see Polya and Read 1987, p.99). They were studied systematically in a series of papers by Kaloujnine and Krasner (1948, 1950, 1951a, 1951b). The Sylow subgroups of symmetric groups (Hall 1959, Section 5.9) are formed from wreath products, as are the Sylow subgroups of Gl(n, Fq) (and other linear groups) away from the characteristic of the field (Weir 1955). The cohomology of wreath products has played an important role in the development of the subject, in part because of interest in the cohomology of such groups, but also because wreath products are closely connected with the notion of transfer (corestriction) and its generalizations. We shall explore these notions in what follows.