On the Commutators of a given Group

we observe that the transform, with respect to any operator of 6r, of the commutators formed with t and the operators of G is the product of two such commutators. All of these commutators must, therefore, form a group which is transformed into itself by G. When t transforms G into itself the given commutators generate the smallest self-conjugate subgroup of G which has the property that all of the operators of the corresponding quotient group are commutative to t ; i. e., t transforms each of the divisions of G with respect to this self-conjugate subgroup into itself. By letting t represent, in succession, all the operators of G we arrive at the known theorem that the commutator subgroup of a group is the smallest self-con jugate subgroup with respect to which the group is isomorphic to an abelian group.* From the fact that the commutator subgroup of G is a characteristic subgroup it follows that it is self-conjugatein every group that contains G self-conjugately. In particular, if G is of order p and is not the abelian group of the type (1, 1, 1, ..., 1), then every group that contains G self-conjugately must also contain a self-conjugate subgroup of order pf*, a > t3 > 0. When G is the abelian group of the type ( 1 , 1 , 1, *.., 1), it is clearly always possible to construct a group which contains G self-conjugately and which transforms none of the subgroups of G besides identity into itself, f If t remains fixed in the commutator G = s ~* t ~ st while « is multiplied on the left by all the operators of G we observe that C remains unchanged when this multiplier is commutative to t and that it is changed for every other multiplier. Arranging all the operators of G in the following