A converse to Lagrange's theorem

Abstract

AbstractAn integer n >0 is called a CLT-number if any group of order n has subgroups of order d for every divisor d of n. The Set of CLT-numbers n is characterized by properties of the prime factorization of n. In addition, if G has order dividing a CLT-number then the structure of G/CG(U) is given where U is a chief factor of G. As a consequence, it is shown that G is solvable of Fitting height at most 3.