ABSTRACT
The uniform structure inside a group (G, *) provided by the set {a * H} of cosets of any of its subgroups H bears particular fruit when G is finite, either Abelian or non-Abelian. This enables to arrive at a fundamentally important result in finite group theory, made possible by this uniformity and known as Lagrange’s Theorem. If we wish to discover a counter-example to possible converse of Lagrange’s Theorem, we need to look in the class of finite non-Abelian groups. The usual focus in that class has been the groups Sn of all permutations of n objects under composition. It turns out that counter-examples exist, but to discover them we need to go into a bit more detail about these permutation groups. The converse of Lagrange’s Theorem is not true in general (but is true for finite Abelian groups).
