A Class of Loops with the Isotopy-Isomorphy Property

A loop L has the isotopy-isomorphy property provided each loop isotopic to L is isomorphic to L. A familiar problem is that of characterizing those loops having this property. It is well known (1, p. 56) that the loop isotopes of (L, ·) are those loops L(a, b, *) defined by x * y = x/b·a\y for some a, b in L. In this paper we first show (Corollary to Theorem 1) that a loop L with identity element 1 has the isotopy-isomorphy property if L is isomorphic to 1,(1, x) and to L(x, 1) for each x in L. We then determine necessary and sufficient conditions (Theorems 2 and 3) for L to be isomorphic to these isotopes under translations (i.e. permutations of the form xv = cx or xv = xc for c fixed).