Commutativity Theorems in Groups with Power-like Maps

Abstract

There are several commutativity theorems in groups and rings which involve power maps f(x) = xⁿ. The most famous example  of this kind is Jacobson's theorem which asserts that any ring satisfying the identity xⁿ = x is commutative.  Such statements belong to first order logic with equality and hence provable, in principle, by any first-order theorem-prover. However, because of the presence of an arbitrary integer parameter n in the exponent, they are outside the scope of any first-order theorem-prover. In particular, one cannot use such an automated reasoning system to prove theorems involving power maps. Here we focus just on the needed properties of power maps f(x) = xⁿ and show how one can avoid having to reason explicitly with integer exponents. Implementing these new equational properties of power maps, we show how a theorem-prover can be a handy tool for quickly proving or confirming the truth of such theorems.