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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] 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 ...
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 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 ...
There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism. The answer is just the dihedral symmetry group Dih n.
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 ...
It should not be confused with the full icosahedral group, which is a different group of order 120, and is rather a subgroup of the orthogonal group O(3). In the algebra of quaternions , the binary icosahedral group is concretely realized as a discrete subgroup of the versors , which are the quaternions of norm one.