The groups in which every two conjugate operators are cominutative have recently been considered by BURNSIDE.t In the first section of the present paper, the converse limitation is imposed on a group of operations. It is assumed that every two commutative operations are conjugate, provided neither is identity, t and the groups which are possible under this hypothesis are determined. It results that the group of order 2 ? and the symmetric group of order 6 are the only grou.ps which have the property in question. In sections 2-6, somewhat similar but smaller limitations are imposed on the group. The condition is imposed in sections 2-5 that every two commutative operations of the same order are conjugate, and in section 6 that every two commutative operations (identity excluded) are so related that each of them is conjugate to some power of the other. Some of the chief properties of the groups which are possible under these limitations are derived. The sections 7-8 deal with problems closely related to the precedilng. If it is assumed that a certain simple relation exists between the number X of complete sets of conjugate operations, and the number n of distinct prime factors in the order of the group, certain commutative operations are conjugate. Much use is made of this fact in showing what groups are possible under the given hypothesis. The symbol Ga (S1= 1), S2, S3, * , * g will be used to represent the group of order g under consideration, and p1, P2, * p Pn to represent distinct primes in ascending order of magnitude so that gp a1pj2. pa-.