On the Invariant Subgroups of Prime Index*

The totality formed by all the operators of any group (G) which are common to all the invariant subgroups of prime index (p) constitutes a characteristic subgroup, and the corresponding quotient group is the abelian group of order pK and of type (1, 1, 1, ■■■)-\ The number of the invariant subgroups of index p is therefore pK — 1/p — 1. The given totality includes all the operators of G which are pt\\ powers, and it is composed of such operators whenever G is abelian. In this case X is clearly equal to the number of the invariants in the Sylow subgroup of order p'" contained in G. This fact follows also directly from the theory of reciprocal groups, since pK is the order of the subgroup generated by all the operators of order p contained in G. For instance, the abelian group G contains only one subgroup of index p whenever its Sylow subgroup of order pm, m > 0, is cyclic, it contains p + 1 such subgroups whenever this Sylow subgroup involves two invariants, p2 + p + 1 whenever there are three invariants, etc. When G is non-abelian the determination of X is much more difficult. Its value can clearly not exceed the value of X for a Sylow subgroup of order pm contained in G, but it may be less. Hence it is of fundamental importance to determine the values of X for the groups of order pTM, and we shall assume that this is the order of G in what follows. Instead of determining the value of X for given types of groups, it seems much more desirable to determine the possible types of groups when the value of X is given. This problem soon becomes very difficult. When X = 1, G is cyclic ; and when X = m, G is the abelian group of type (1, 1, 1, ■••). These extreme cases relate to fundamental groups whose elementary properties are well known. The case when X = m — 1 includes the Hamiltonian groups. All the groups which belong to this case have recently been determined. $ The main object of the present paper is the study of an interesting category of groups which belong to the case when