On the Concept of Co-Sets in a Semi-Group.

and a group of type ln it cannot have the property that each of its subgroups of composite order is invariant. If the order of G is not a power of a prime number each of its Sylow subgroups whose order exceeds a prime number is invariant under G and hence G involves invariantly the direct product of all its Sylow subgroups of composite orders. The quotient group of G with respect to this direct product has an order which is the product of distinct prime numbers. In particular, it results that whenever every subgroup of composite order contained in a group is invariant then the group is solvable. If the given quotient group involves a non-invariant operator of prime order which is not equal to the smallest prime number which divides the order of this quotient group then this operator is non-commutative with an operator of smallest order in this quotient group and this pair of orders is the only pair of prime divisors of the order of this quotient group such that these divisors are the orders of two non-commutative operators of this quotient group. This fact results directly from the condition that every subgroup of composite order contained in G is invariant under G. Hence it results that when this quotient group is non-cyclic it is the direct product of a cyclic group and a group whose order is the product of two prime numbers, one of which is the smallest prime number which divides the order of this quotient group.