1. In my investigations in group theory, I have observed that Lagrange's theorem (that the order of a group is divisible by the order of any subgroup) does not use for its proof the Associative Law in its whole extent; this law can be replaced by a more general postulate, "Postulate A", as I shall call it. We shall represent our elements by capital italic letters; the operation upon them may be represented by a star *, so that A*B signifies the result of this operation performed upon A and B. A set of elements closed under any operation * may be called "a group"; this word is thus used in a more general sense than is usual, since the operation * is arbitrary. The ordinary groups with a special well known operation may be called "classic" to distinguish them from our generalised groups. Sets and groups will be denoted by capital German letters. POSTULATE A. In the equation