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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 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 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 ...
Vito the pug, the winner of the Toy Group, has been crowned Best in Show at the 2024 National Dog Show. Handler Michael Scott called Vito "a very compact dog" as he accepted the trophy.
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 ...
The elements x and y generate a normal subgroup isomorphic to the quaternion group of order 8. The center is cyclic of order 2m. It is generated by the elements z 3 and x 2 = y 2, and the quotient by the center is the tetrahedral group, equivalently, the alternating group A 4. When m = 1 this group is the binary tetrahedral group.