The prove for this property is not hard, since the number of irreducible characters of the quaternion group equals to the number of conjugacy classes of the quaternion group, which is five ( { e }, { }, { i, }, { j, }, { k, } ).
42.
The generalized quaternion group, the dihedral group, and the quasidihedral group of order 2 " n " all have nilpotency class " n " " 1, and are the only isomorphism classes of groups of order 2 " n " with nilpotency class " n " " 1.
43.
It " shows how various physical covariance groups : SO ( 3 ), the Lorentz group, the general relativity group, the Clifford algebra SU ( 2 ), and the conformal group can be readily related to the quaternion group " in Alexander Macfarlane and his " Bulletin " of the Quaternion Society.
44.
The smallest abstract groups that are " not " any symmetry group in 3D, are the quaternion group ( of order 8 ), Z 3 & times; Z 3 ( of order 9 ), the dicyclic group Dic 3 ( of order 12 ), and 10 of the 14 groups of order 16.
