Group cohomology

introduction

• For a finite group G and its module M, \(H^{0}(G,M)\) is isomorphic to \(M/N(M)\) where \(N(m) = \sum_{\sigma\in G}\sigma m\)

central extension

• A central extension of a group \(G\) is a short exact sequence of groups

\[1\rightarrow A\rightarrow E\rightarrow G\rightarrow 1\] such that \(A\) is in \(Z(E)\), the center of the group \(E\).

• The set of isomorphism classes of central extensions of \(G\) by \(A\) (where \(G\) acts trivially on \(A\)) is in one-to-one correspondence with the cohomology group \(H^2(G, A)\).

expositions

• Sprehn, David. 2014. “Nonvanishing Cohomology Classes on Finite Groups of Lie Type with Coxeter Number at Most P.” arXiv:1407.3299 [math], July. http://arxiv.org/abs/1407.3299.
• Nakano, Daniel K. “Cohomology of Algebraic Groups, Finite Groups, and Lie Algebras: Interactions and Connections.” arXiv:1404.3342 [math], April 12, 2014. http://arxiv.org/abs/1404.3342.
• Alejandro Adem, Recent developments in the cohomology of finite groups, Notices Amer. Math. Soc. 44 (1997), no. 7, 806–812 http://www.ams.org/notices/199707/adem.pdf