-constraint of the automorphism group of a finite group
We investigate loops defined upon the product by the formula , where , for appropriate parameters . Each such loop is coupled to a 2-cocycle (in the group-theoretical sense) and this connection makes it possible to prove that the loop possesses a metacyclic inner mapping group. If , then the loop is an A-loop. Questions of isotopism and isomorphism are considered in detail.
We report on a partial solution of the conjecture that the class of finite solvable groups can be characterised by 2-variable identities. The proof requires pieces from number theory, algebraic geometry, singularity theory and computer algebra. The computations were carried out using the computer algebra system SINGULAR.