A direct proof of Leutbecher's Lemma

Extract

Using the theory of group extensions, A. Leutbecher [1] proved this

Lemma. Let G bea group, and w some 2-cocycle of a trivial G-module M. The cohomology class ofw will contain symmetric cocycles if and only if w is semisymmetric.

Here we have called w symmetric or semisymmetric according as w(h, g) = w(g, h) for all g, h ∈G or only for those with hg = gh. In one direction, the proof reduces to observing that 2-coboundaries of trivial G-modules are semisymmetric. The nontrivial part of the lemma also admits of a straightforward proof, as follows.