Zorn's matrices and finite index subloops

Abstract

The Zorn's Algebra ZZ(R) has a multiplicative function called determinant with properties similar to the usual one. The set of elements in ZZ(R) with determinant 1 is a Moufang loop that we will denote by \GA. In our main result we prove that if R is a Dedekind algebraic number domain that contains an infinite order unit, each finite index subloop L, such that \GA has the weak Lagrange property relative to L, is congruence subloop. In addition, if R=\Z, then we present normal subloops of finite index in \GA that are not congruence subloops.