In group theory, the quaternion group is a nonabelian group of order 8, isomorphic to a certain eightelement subset of the quaternions under multiplication. It is often denoted by Q or Q_{8}, and is given by the group presentation
where 1 is the identity element and −1 commutes with the other elements of the group.
The Cayley table (multiplication table) for Q is given by^{[1]}:
The multiplication of pairs of elements from the subset {±i, ±j, ±k} works like the cross product of unit vectors in threedimensional Euclidean space.
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Properties
The quaternion group has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is nonabelian.^{[2]} Every Hamiltonian group contains a copy of Q.^{[3]}
In abstract algebra, one can construct a real 4dimensional vector space with basis {1, i, j, k} and turn it into an associative algebra by using the above multiplication table and distributivity. The result is a skew field called the quaternions. Note that this is not quite the group algebra on Q (which would be 8dimensional). Conversely, one can start with the quaternions and define the quaternion group as the multiplicative subgroup consisting of the eight elements {1, −1, i, −i, j, −j, k, −k}. The complex 4dimensional vector space on the same basis is called the algebra of biquaternions.
Note that i, j, and k all have order 4 in Q and any two of them generate the entire group. Another presentation of Q^{[4]} demonstrating this is:
One may take, for instance, i = x, j = y and k = xy.
The center and the commutator subgroup of Q is the subgroup {±1}. The factor group Q/{±1} is isomorphic to the Klein fourgroup V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein fourgroup. The full automorphism group of Q is isomorphic to S_{4}, the symmetric group on four letters. The outer automorphism group of Q is then S_{4}/V which is isomorphic to S_{3}.
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