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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]
Thus the multiplicative group of nonzero quaternions acts by conjugation on the copy of consisting of quaternions with real part equal to zero. Conjugation by a unit quaternion (a quaternion of absolute value 1) with real part cos( φ ) is a rotation by an angle 2 φ , the axis of the rotation being the direction of the vector part.
In mathematics, a quaternionic structure or Q-structure is an axiomatic system that abstracts the concept of a quaternion algebra over a field.. A quaternionic structure is a triple (G, Q, q) where G is an elementary abelian group of exponent 2 with a distinguished element −1, Q is a pointed set with distinguished element 1, and q is a symmetric surjection G×G → Q satisfying axioms
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 ...
The World Canine Federation recognizes 350 unique dog breeds. In the U.S. The American Kennel Club now recognizes 209 breeds. That’s…a lot of dogs. To better understand each breed, humans have ...
From this point of view, quaternionic representation of a group G is a group homomorphism φ: G → GL(V, H), the group of invertible quaternion-linear transformations of V. In particular, a quaternionic matrix representation of g assigns a square matrix of quaternions ρ(g) to each element g of G such that ρ(e) is the identity matrix and
The quaternion group has normal subgroups of order 4 and 2 but is not a [semi]direct product. Schur's papers at the beginning of the 20th century introduced the notion of central extension to address examples such as C 4 {\displaystyle C_{4}} and the quaternions.