The third cohomology group of a monoid and admissible abstract kernels

We define the product of admissible abstract kernels of the form φ: M → End(G)/Inn(G), where M is a monoid, G is a group and φ is a monoid homomorphism. Identifying C-equivalent abstract kernels, where C is the center of G, we obtain that the set M(M,C) of C-equivalence classes of admissible abstract kernels inducing the same action of M on C is a commutative monoid. Considering the submonoid L(M,C) of abstract kernels that are induced by special Schreier extensions, we prove that the factor monoid (M,C) = M(M,C) L(M,C) is an abelian group. Moreover, we show that this abelian group is isomorphic to the third cohomology group H3(M,C).