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The quaternion group has the unusual property of being Hamiltonian: Q 8 is non-abelian, but every subgroup is normal. [4] Every Hamiltonian group contains a copy of Q 8. [5] The quaternion group Q 8 and the dihedral group D 4 are the two smallest examples of a nilpotent non-abelian group.
The real quaternion 1 is the identity element. The real quaternions commute with all other quaternions, that is aq = qa for every quaternion q and every real quaternion a. In algebraic terminology this is to say that the field of real quaternions are the center of this quaternion algebra.
For even n ≥ 4, the center consists of the identity element together with the 180° rotation of the polygon. The center of the quaternion group, Q 8 = {1, −1, i, −i, j, −j, k, −k}, is {1, −1}. The center of the symmetric group, S n, is trivial for n ≥ 3. The center of the alternating group, A n, is trivial for n ≥ 4.
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
Comment: The proof of Euler's four-square identity is by simple algebraic evaluation. Quaternions derive from the four-square identity, which can be written as the product of two inner products of 4-dimensional vectors, yielding again an inner product of 4-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b).
3D visualization of a sphere and a rotation about an Euler axis (^) by an angle of In 3-dimensional space, according to Euler's rotation theorem, any rotation or sequence of rotations of a rigid body or coordinate system about a fixed point is equivalent to a single rotation by a given angle about a fixed axis (called the Euler axis) that runs through the fixed point. [6]
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 ...
Various cultural studies and social theory investigate the question of cultural and ethnic identities. Cultural identity adheres to location, gender, race, history, nationality, sexual orientation, religious beliefs, and ethnicity. [22] National identity is an ethical and philosophical concept where all humans are divided into groups called ...
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