The generalized quaternion groups have the property that every " p "-group with this property ( every abelian subgroup is cyclic ) is either cyclic or a generalized quaternion group as defined above.
32.
The generalized quaternion groups have the property that every " p "-group with this property ( every abelian subgroup is cyclic ) is either cyclic or a generalized quaternion group as defined above.
33.
The group of units in " L " is the order 8 quaternion group The group of units in " H " is a nonabelian group of order 24 known as the binary tetrahedral group.
34.
While no one knows exactly why, there seems to be a four-year cycle in the economy and the stock market, according to Marc M . Groz, president of the Quaternion Group, a research firm in New York.
35.
Conversely, one can start with the quaternions and " define " the quaternion group as the multiplicative subgroup consisting of the eight elements The complex four-dimensional vector space on the same basis is called the algebra of biquaternions.
36.
The character table does not in general determine the group up to isomorphism : for example, the quaternion group " Q " and the dihedral group of 8 elements ( " D " 4 ) have the same character table.
37.
In particular, for a finite field " F " with odd characteristic, the 2-Sylow subgroup of SL 2 ( " F " ) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group,.
38.
The tame blocks ( which only occur for the prime 2 ) have as a defect group a dihedral group, semidihedral group or ( generalized ) quaternion group, and their structure has been broadly determined in a series of papers by Karin Erdmann.
39.
It is relatively trivial to prove that groups with different identity skeletons cannot be isomorphic, though the converse is not true ( for instance, the cyclic group " C 8 " and the quaternion group " Q " are non-isomorphic but have the same identity skeleton ).
40.
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, } ).
