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Recent Developments in
the Cohomology of
Finite Groups
Alejandro Adem
In this expository paper we describe some recent of these functions in a functorial way. In par-
work in the cohomology of finite groups. In par- ticular, isomorphism classes of extensions as de-
ticular, we discuss techniques for calculation and scribed above are in one-to-one correspondence
how they apply to key examples from group the- with these equivalence classes of 2-cocycles,
ory, namely, the sporadic simple groups. We also which can be regarded as elements in a “coho-
mention recent theoretical results arising from mology group”, denoted by H2(K,N) (see [3],
these calculations and interactions between al- Chapter I). It was, however, the impetus from
gebra and topology which occur in the context of topology that eventually gave rise to a global de-
finite group cohomology.
finition of group cohomology using machinery
Definitions and Motivation from homological algebra, generalizing specific
low-dimensional information. We refer the reader
Let G denote a finite group. One of the basic
tools for understanding G is its expression as to the excellent historical account by S. Mac Lane
a group extension [17].
Let us review the essential elements in the gen-
→→→→
1 N G K 1 eral definition (plete details the reader
Z
where N is