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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]
Coren's book presents a ranked list of breed intelligence, based on a survey of 208 dog obedience judges across North America. [10] When it was first published there was much media attention and commentary in terms of both pros [11] and cons. [12] Over the years, Coren's ranking of breeds and methodology have come to be accepted as a valid description of the differences among dog breeds in ...
Dog intelligence or dog cognition is the process in dogs of acquiring information and conceptual skills, and storing them in memory, retrieving, combining and comparing them, and using them in new situations. [1] Studies have shown that dogs display many behaviors associated with intelligence. They have advanced memory skills, and are able to ...
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 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 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 ...
The quaternion group Q 8 of order 8, which has 6 elements of order 4. If n is a positive integer there are two extraspecial groups of order 2 1+2n, which are given by The central product of n extraspecial groups of order 8, an odd number of which are quaternion groups. The corresponding quadratic form (see below) has Arf invariant 1.