On the classification of just-non-Cross varieties of groups

Abstract

Apart from some (insoluble) subvarieties of , the jnC (just-non-Cross) varieties known so far comprise the following list: , , , , where p, q and r are any three distinct primes. In a recent paper I gave a partial confirmation of the conjecture that the soluble jnC varieties all appear in this list. Here I show that a jnC variety is reducible if and only if it is soluble of finite exponent; this reduces the problem of classifying jnC varieties to finding the irreducibles of finite exponent. I observe that these fall into three distinct classes, and show that the questions of whether or not two of these classes are empty have some bearing on some apparently difficult problems of group theory.