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.
Various commutators and associators may be defined in one-sided loops. In this paper, we approximate and compare these objects in the left and right loop reducts of a Catalan loop. To within a certain order of approximation, they turn out to be quite symmetrical. Using the general analysis of commutators and associators, we investigate the structure of a specific Catalan loop which is non-commutative, but associative, that appears in the original number-theoretic application of Catalan loops.
In this short note, it is shown that if are -connected transversals for a finite subgroup of an infinite group such that the index of in is at least 3 and whenever and then is a normal abelian subgroup of .
Scriviamo ed . Cerchiamo gruppi con generatori tali che ed per alcuni numeri naturali , .