MacLane method of group cohomologies and gauge theories on finite lattices
It is well known that the set of inequivalent extensions of a group \(Q\) by an Abelian group \(T\), with given action of \(Q\) on \(T\), can be labelled by the elements of a second cohomology group. It is also well known that the classification of crystallographic space groups falls within this extension framework, as does the so-called magnetic translation group in the Brown-Zak theory of an electron moving in a crystal under the additional influence of a uniform magnetic field. A theoretical labelling is one thing, but the actual computation of the second cohomology group is not usually that easy. The MacLane free group method offers a considerable reduction in the combinatorial explosion inherent in any direct attack on the problem. This method is reviewed, as are the two applications referred to above.
computations of second cohomology groups
scientific article; zbMATH DE number 639206
scientific article