FREE GROUPS AND UNIFICATION IN U_mU_2

摘要

The problem of solving equations, with or without parameters, in the absolutely free group has a lengthy history. Various results around this area are discussed in the last few sections of the first chapter of [6]. John Lawrence [5] introduced us to the problem restricted to other varieties of groups. In these notes he has a number of results which indicate that "most" varieties generated by a finite group will have equations for which there are no most general solutions. The fact that a case which remained open from his work was "non-square free exponent and abelian Sylow subgroups" led us to our consideration of the dihedral groups. It is not hard (but of doubtful utility) to generalize the results we have to the case of varieties U_mU_n where n|(q - 1) for every prime divisor q of m (in this case Z_m contains a non-trivial nth root of unity and the "diagonaliza-tion"can still be carried out). For the general case U_mU_n when m and n are relatively prime, it seems that the main result (existence of most general solutions) is still correct. However, the direct sum decomposition which figures so prominently in the argument is no longer so easily described, and such technical obstructions have prevented us from actually describing the solutions in this case.