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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 quaternion group, Q8= 2,2,2 , consisting of the 8 Lipschitz units forms a subgroup of index 15, which is also the dicyclic group Dic 2; this covers the stabilizer of an edge. The 24 Hurwitz units form an index 5 subgroup called the binary tetrahedral group ; this covers a chiral tetrahedral group .
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 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 ...
The group of units in L is the order 8 quaternion group Q = {±1, ±i, ±j, ±k}. The group of units in H is a nonabelian group of order 24 known as the binary tetrahedral group. The elements of this group include the 8 elements of Q along with the 16 quaternions {(±1 ± i ± j ± k) / 2}, where signs may be taken in any combination. 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 ...
In the quaternion group of order 8, each of the cyclic subgroups of order 4 is normal, but none of these are characteristic. However, the subgroup, {1, −1}, is characteristic, since it is the only subgroup of order 2. If n > 2 is even, the dihedral group of order 2n has 3 subgroups of index 2, all of which are normal. One of these is the ...