Cycle diagram of Q. Each color specifies a series of powers of any element connected to the identity element (1). For example, the cycle in red reflects the fact that i ² = −1, i ³ = −i  and i ⁴ = 1. The red cycle also reflects the fact that (−i )² = −1, (−i )³ = i  and (−i )⁴ = 1.

In group theory, the quaternion group is a non-abelian group of order 8 with a number of interesting properties. The quaternion group, often denoted by Q, is usually written in multiplicative form, with the following 8 elements Cycle diagram of the Q8 group File links The following pages link to this file: Quaternion group Cycle graph (group) Categories: User-created public domain images ... In group theory a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. ... Group theory is that branch of mathematics concerned with the study of groups. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ...

Q = {1, −1, i, −i, j, −j, k, −k}

Here 1 is the identity element, (−1)² = 1, and (−1)a = a(−1) = −a for all a in Q. The remaining multiplication rules can be obtained from the following relation:

i² = j² = k² = ijk = − 1

The entire Cayley table (multiplication table) for Q is given by: A Cayley table is a representation of a product defined on a set G. It is a group-theoretic generalization of an addition or a multiplication table. ...

Note that the resulting group is non-commutative; for example ij = −ji. Q has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian. Every Hamiltonian group contains a copy of Q. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... In group theory, a non-abelian group G is called Hamiltonian if every subgroup of G is normal. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are...

In abstract algebra, one can construct a real 4-dimensional 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 8-dimensional). 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}. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, an associative algebra is a vector space (or more generally module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... In abstract algebra, a division ring, also called a skew field, is a ring with 0 ≠ 1 and such that every non-zero element a has a multiplicative inverse (i. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ...

Note that i, j, and k all have order 4 in Q and any two of them generate the entire group. Q has the presentation In mathematics, one method of defining a group is by a presentation. ...

One may take, for instance, x = i and y = j.

The center of Q is the subgroup {±1}. The factor group Q/{±1} is isomorphic to the Klein four-group V. The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. The full automorphism group of Q is isomorphic to S₄, the symmetric group on four letters. The outer automorphism group of Q is then S₄/V which is isomorphic to S₃. In abstract algebra, the center (or centre) of a group G is the set Z(G) of all elements in G which commute with all the elements of G. Specifically, Z(G) = {z ∈ G | gz = zg for all g ∈ G} Note that Z(G) is a subgroup of G — if... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V), named after Felix Klein, is the group C2 × C2, the direct product of two copies of the cyclic group of order 2 (or any isomorphic variant). ... In abstract algebra, an inner automorphism of a group is a function f : G -> G defined by f(x) = axa-1     for all x in G; where the conjugation is often denoted exponentially by xa. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In mathematics, the outer automorphism group of a group G is the quotient of the automorphism group Aut(G) by its inner automorphism group Inn(G). ...

The quaternion group Q may be regarded as acting on the eight nonzero elements of the 2-dimensional vector space over the finite field GF(3). For a picture, see Visualizing GL(2,p). In abstract algebra, a finite field or Galois field (so named in honor of Evariste Galois) is a field that contains only finitely many elements. ...

Generalized quaternion group

A group is called a generalized quaternion group if it has a presentation In mathematics, one method of defining a group is by a presentation. ...

for some integer n ≥ 3. The order of this group is 2ⁿ. The ordinary quaternion group corresponds to the case n = 3. The generalized quaternion group can be realized as the subgroup of unit quaternions generated by

The generalized quaternion groups are members of the still larger family of dicyclic groups. The generalized quaternion groups have the property that every abelian subgroup is cyclic. 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. In group theory, a dicyclic group is a member of a class of groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name di-cyclic). ... In mathematics, an abelian group is a commutative group, i. ... In mathematics, given a prime number p, a p-group is a group in which each element has a power of p as its order. ...

See also

• quaternion
• Klein four-group
• dicyclic group
• Hurwitz integral quaternion
• 16-cell