"J898.J MINUTES . 319 



While the direct product of the quaternion group and any Abelian 

 group of an odd order is always a Hamiltonian group, the direct 

 product of the quaternion group and an Abelian group whose order 

 is divisible by a power of 2 is only Hamiltonian when the latter 

 group contains no operator whose order is divisible 4. This follows 

 directly from the fact that the group generated by the product of 

 an operator of order 4 in the Hamiltonian group and any operator 

 in such an Abelian group must be self- conjugate. 



We may determine the number of the quaternion groups that are 

 contained in a Hamiltonian group whose order is divisible by 2' 

 without being divisible by 2*"^^ in the following manner. Such a 

 group contains a single subgroup ^ of order 2". This subgroup in- 

 cludes 3 times 2"~^ operators of order 4. Each quaternion subgroup 

 includes two of the operators of order 4 that are included in a sub- 

 group of order 2"".^ which involves only 2""^ operators of order 4. 

 Hence there are 2^' * quaternion subgroups in the given Hamiltonian 

 group. All of these have the commutator subgroup of the entire 

 group in common. In other words, the commutator subgroup of a 

 Hamiltonian group is the same as that of any one of its quaternion 

 ■subgroups. 



Cornell University, June, 1898, 



Stated Meeting, October ?>1, 1898. 



Vice-President Sellers in the Chair. 



Present, 12 members. 



Prof. Lighter Witmer, a newly elected m^ember, was pre- 

 f;ented to the Chair, and took his seat. 



The minutes of the last stated meeting were read and 

 approved . 



Dr. Frazer read a letter from the International Geological 

 Congress in regard to the establishment of an international 

 floating institute, and offered the following resolution : 



Resolved, That the President of the Society be requested to 

 memorialize Congress in favor of an appropriation in aid of the in- 



* Sylow, Mathematische AnnaUn, 1872, Vol. v, p. 584. 

 PROC. AMEB. PHILOS, SOC. XXXVII. 158. U. PRINTED FEB. 23, 1899. 



