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It can be shown that a finite p-group with this property (every abelian subgroup is cyclic) is either cyclic or a generalized quaternion group as defined above. [12] Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or a 2-group isomorphic to generalized quaternion group. [13]
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The quaternions are "essentially" the only (non-trivial) central simple algebra (CSA) over the real numbers, in the sense that every CSA over the real numbers is Brauer equivalent to either the real numbers or the quaternions. Explicitly, the Brauer group of the real numbers consists of two classes, represented by the real numbers and the ...
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. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a ...
Reverted to version as of 15:30, 26 June 2013 -- more recent upload messed up margins and several other things, and I really can't tell what the improvement is 16:45, 17 July 2013 256 × 161 (9 KB)
The seven major dog groups in the U.S. are Herding, Hound, Non-Sporting, Sporting, Terrier, Toy and Working. Initially, when the AKC got its start in 1884, it tossed all dog breeds into either the ...
The most familiar (and smallest) example of a Hamiltonian group is the quaternion group of order 8, denoted by Q 8. Dedekind and Baer have shown (in the finite and respectively infinite order case) that every Hamiltonian group is a direct product of the form G = Q 8 × B × D , where B is an elementary abelian 2-group , and D is a torsion ...
Cayley Q8 graph of quaternion multiplication showing cycles of multiplication of i (red), j (green) and k (blue). In the SVG file, hover over or click a path to highlight it. All of the Clifford algebras Cl p , q ( R {\displaystyle \mathbb {R} } ) apart from the real numbers, complex numbers and the quaternions contain non-real elements that ...