The quaternion group is usually written in multiplicative form, with the following 8 elements
i  j  k  
i  1  k  j 
j  k  1  i 
k  j  i  1 
Note that the resulting group is noncommutative; for example ij = ji.
Q_{8} has the unusual property of being Hamiltonian: every subgroup of Q_{8} is a normal subgroup, but the group is nonabelian. Every Hamiltonian group contains a copy of Q_{8}.
In abstract algebra, we 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.
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}.
Q_{8} has a presentation with generators {x,y} and relations x^{4} = 1, x^{2} = y^{2}, and y^{1}xy = x^{1}. (For example x = i, y = j.) A group is called a generalized quaternion group if it has a presentation, for some integer n > 1, with generators {x,y} and relations x^{2n} = 1, x^{2n1} = y^{2}, and y^{1}xy = x^{1}. These groups are members of the still larger family of dicyclic groups.
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