Formal power series representations of free exponential groups

摘要

A classical result of Magnus asserts that a free groupFhas a faithful representation in the group of units of a ring of non-commuting formal power series with integral coefficients. We generalize this result to the category of A-groups, where A is an associative ring or an Abelian group. We say that a free A-group FAhas a faithful Magnus representation if there is a ring B containing A as an additive subgroup (or a subring) such that FAis faithfully represented (exactly as in Magnus' classical result in the case A = Z)in the group of units of the ring of formal power series in non-communting indeterminater over B.The three principal results are: (I) If A is a torsion free Abelian group and FAis a free A-groupp of Lyndon' type, then FAhas a faithful Magnus representation; (II) If A is an ω‐residually Z ring, then FAhas a faithful Magnus representation;(III) for every nontrivial torsion-free Abelian group A, FAhas a faithful Magnus representation in B[[Y]] for a suitable ring B in and only if FQhas a faithful Magnus representation in Q[[Y]].