= It was quite possible to ascertain Ae= 0.1; i.e., elongations of but Al/l= 5 × 10-8, equivalent to Al 2.3 X 10-5 cm. The current must exceed 0.02 am. before any elongation can be detected, after which, however, the elongations abruptly begin and increase rapidly to a maximum, which is reached before saturation. The experiments, figure 2, were made with somewhat greater care and with larger fringes. The standardization of the ocular micrometer showed But for incidental difficulties (tremors, etc,), the results in figure 2 would probably be very smooth. = A number of supplementary experiments (see figure 2) were made to see whether the observed Al 0 for currents below 0.02 amperes might not be equivalent to an initial small minimum. But Al remained persistently zero, while currents decreased from 0.02 to 0.001 amperes. At 0.004 am. the field was reversed, but no significant Al could be observed. The fringes just moved (Ae= 0.1) when i was about 0.035 amperes, indicating a field of 3 or 4 gauss. A rough test made of the equation by pushing the rod rr forward by the backstop screw M, figure 1, gave corroborative results. If A refers to the turns of the screw M (AN/Ae) was found to be 10-5 X 3.3 cm. per scale part and (Ae/Ap) scale parts per degree of turn. Hence with the above data Al = 104 X 1.8 The back screw having 40 turns to the inch, i.e., a pitch of 0.0635 cm. gives us 104 X 1.76 cm. per degree of turn. Another feature may be mentioned here. The expansion of the coil when carrying very large currents is a thrust on the back stop M, which is quite appreciable and appears as an apparent contraction of the rod. 4. Vibration telescope.-To test the surmise that initial elongations always precede the final contraction, the vibration telescope heretofore described1 was installed. It was then found that the even band of fringes drawn out by the vibrating objective broke up into strongly sinuous lines on making and particularly on breaking the circuit through the helix. When the circuit was made and broken alternately, the waves broke up further into a succession of discontinuous pulses of more than double the amplitudes of the waves. With the field properly adjusted by passing 1.8 to 2 amperes through the coil, there was no further observable displacement after the strong waves lines, produced immediately after closing the circuit, had subsided. 5. Further observations.-The data of figure 3 were investigated in a single series. To reduce the heat discrepancy a brisk current of water was passed through the tubular water jacket. This seemed the safer plan, even though the fringes were shaken. The observations were made in triplets and largely confined to the higher fields. The curve is quite as clearly indicated as may be expected owing to the difficulties cited; but the higher observations (H>200) are decreasing contractions. The reason of this is partly owing to the method of observation in triplets, where (curiously enough) the third reading (field zero) was a contraction in relation to the first reading in the absence of the field. The apparatus in these experiments was therefore suspected of being faulty in design, inasmuch as the clutch of the contact lever and of the coil were attached to the same rigid standard. This arrangement was now modified, so that the two mountings were quite independent, whereupon the anomalous results specified largely receded. As a second test a rod 28 cm. long of Swedish iron was inserted, the extra length being pieced out by brass tubing soldered to each end, so that the iron lay quite within the coil. The data obtained closely resembled figure 3. Tests made with other metals gave no positive results. 6. Coefficient of expansion.—To arrive at a definite reason for the occurrence of the anomalous contraction mentioned above, it seemed desirable to modify the magnetic apparatus for the purpose of measuring the coefficient of expansion of a given metal. This could easily be done by using the coil merely as a heater. A number of experiments were made, using either the ocular micrometer (here the temperature increments must lie within 2°) and the Fraunhofer micrometer at the mirror of the interferometer. It was observed that all the expansions were apt to begin with a contraction immediately after the heating current had been closed. Hence there is an initial expansion of the coil itself. It was soon found that the consequent flexure of the table was the ultimate cause of the interferometer discrepancy. A modification of the apparatus was made therefore by allowing the end C of the coil to recline on a large grooved wheel, which by slight rotation would admit of any expansion of the kind in question. With this improvement the anomalous contractions vanished and the work thereafter proceded smoothly. 7. Theoretical observations.-To account for such a graph as figure 3, as a whole, one may argue that the initial elongations are to be referred to the rotations of the molecular magnets. For these elongations are coextensive in field variations with the marked increase of magnetization. It would then seem plausible that thereafter the attractions between the oriented molecular magnets may be instanced to account for the persistent contractions in continually increasing fields. Thus it seems worth while to endeavor to ascertain whether such a supposition would conform with any reasonable value of the susceptibility k of the iron, which one may estimate as decreasing from over 100 to less than 10 as the rod approaches saturation (H= 150 gauss) and to decrease thereafter asymptotically to zero. If p is the force per square centimeter of section of the rod and E Young's modulus, p = E(Al/l) regarding the magnetic stress as traction. (1) Using the expression for the potential of a disc, the field F in a narrow crevasse normal to F, between molecular layers of magnetic surface density of magnetization kH where k is the susceptibility of the metal. Hence the force per square centimeter should be p' = FkH, or If the data in figure 3 are taken above 800 gauss, supposing that these are far enough removed from the initial complications, the estimate would be (E = 2 x 1012), k = 1.6. An order of mean susceptibility of 1.6 (which seems not an unreasonable assumption) would thus account for the observed contractions. Naturally as k is essentially variable with H a better statement of the case might be given by postulating such a relation. 1 London, Phil. Mag., 37, 1894, (131). 2 Carnegie Inst., Washington, Pub., No. 149. 3 These PROCEEDINGS, 5, 1919, (39). These PROCEEDINGS, 4, 1918, (328). GROUPS INVOLVING ONLY TWO OPERATORS WHICH ARE SQUARES By A. G. MILLER DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS Communicated by E. H. Moore, May 2, 1919 .) is completely The abelian group of order 2" and of type (1, 1, 1, . characterized by the fact that all of it operators have a common square. When we impose the condition that the operators of a finite group G have two and only two distinct squares then G must belong to one of three infinite systems of groups whose characteristic properties we proceed to determine. All the operators of G besides the identity must be of order 2 or of order 4 and G must involve operators of each of these two orders. Hence the order of G is of the form 2". When G is abelian it is of type (2, 1, 1, . . .) and it will therefore be assumed in what follows that G is non-abelian. The octic group and the quaternion group constitute well known illustrations of such a group and have the smallest possible order. When the operators of order 2 contained in G together with the identity constitute a subgroup this subgroup is the central of G and hence G belongs to the system of groups called Hamiltonian by R. Dedekind. In this case it is known that G is the direct product of the quaternion group and an abelian group of order 2a and of type (1, 1, 1, . . . ). Hence it will be assumed in what follows that G involves non-commutative operators of order 2. Every operator of order 4 contained in G is transformed either into itself or into its inverse by every operator of G and an operator of order 2 contained in G has at most two conjugates under the group. Let H1, H2 represent subgroups composed respectively of all the operators of G which are commutative with two non-commutative operators of order 2 S1, S2. The cross-cut K1 of H1 and H2 is of index 4 under G and includes the central of G. A set of independent generators of G can be so selected as to include S1, S2 and operators from K1. 1 Exactly one-half of the operators of G which are not also in K1, are of order 2 since the quotient group G/K1 is abelian. If K1 involves non-commutative operators of order 2 two such operators S3, s, may be selected from K1 in exactly the same way as s1 and s2 were selected from G. The remaining operators of a set of independent generators including S1, S2, S3, S4 may be selected from an invariant subgroup of index 4 under K, and of index 16 under G all of whose operators are commutative with each of the four operators already chosen. . . As G is supposed to be of finite order we arrive by this process at a subgroup K, in which all the operators of order 2 are commutative. Hence Km belongs to one of the following three well known categories of groups. Abelian and of type (1, 1, 1, ), abelian and of type (2, 1, 1, . . .), or Hamiltonian of order 2a. The commutator subgroup of G is of order 2. In each case, G may be constructed by starting with Km, forming the direct product of K, and an operator t of order 2, and then extending this direct product by means of an operator t2 of order 2 which is commutative with each of the operators of K, and transforms t1 into itself multiplied by the commutator of order 2 contained in G. When K, is Hamiltonian or abelian and of type (2, 1, 1, . . . ) this commutator is determined by K. In the other possible case it may be selected arbitrarily from the operators of order 2 found in Km. When m>1, we use the group K-1 just constructed in exactly the same way as K, was used in the preceding paragraph. The commutator of order 2 is completely determined for each of the categories by K-1, m>1. When m>2 we proceed in the same manner with K-2, etc. It may be noted that in each of the groups belonging to one of the three categories thus constructed more than one-half of the operators are of order 2, in those belonging to the second category the number of operators of order 2 is one less than one-half of the order of G, while in those belonging to the third category the number of operators of order 2 is obtained by subtracting from one-half the order of G one plus one-fourth the order of Km. Some of these results constitute a proof of the following theorem: If only two of the operators of a group G are the squares of operators contained in G then the non-invariant operators of G have only two conjugates, each cyclic subgroup of order 4 is invariant, and G belongs to one of three categories of groups of order 2 which can be separately generated by a set of operators such that each of these operators is commutative with each of the others except at most one of them. When m is sufficiently large there is one and only one group belonging to each of these three categories and having a give number y of pairs of noncommutative operators of order 2 in its set of independent generators when this set is obtained in the manner described above. The smallest values of m for these categories are 2y + 1, 2y + 2, and 2y + 3 respectively. When m has a larger value G must be the direct product of an abelian group of type (1, 1, 1, . . . ) and of the minimal group having y such pairs of generators and contained in the category to which G belongs. By means of these facts it is very easy to determine the number of the groups of a given order 2" which belong to each of these three categories. This number is the largest integer which does not exceed m 2 1 m 2 m 3 and " 2 2 for the three categories respectively. In particular, the number of the distinct groups of order 128 belonging to each of these categories is 3, 2, 2 respectively, it being assumed that each of the groups in question contains at least two non-commutative operators of order 2. In each one of these groups every two non-commutative operators of order 2 generate the octic group and every two non-commutative operators of order 4 generate the quaternion group. Moreover, every non-abelian subgroup is invariant. In two of the categories the central is composed of operators of order 2 in addition to the identity, while the central of the remaining category is of type (2, 1, 1, . . . ). Every one of these groups is generated by its operators of order 2. From the standpoint of definition and structure these categories rank among the simplest known infinite systems of non-abelian groups 1 Dedekind, R., Math. Ann., Leipzig, 48, 1897, (548–561). 2 Miller, G. A., Trans. Amer. Math. Soc., New York, 8, 1907, (1–13). |