Quaternion group - Citizendia

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.

Cycle diagram of Q. In Group theory, a sub-field of Abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful 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. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element It is often denoted by Q and written in multiplicative form, with the following 8 elements

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:

i 2 = j 2 = k 2 = i j k = - 1

The entire Cayley table (multiplication table) for Q is given by:

	1	−1	i	−i	j	−j	k	−k
1	1	−1	i	−i	j	−j	k	−k
−1	−1	1	−i	i	−j	j	−k	k
i	i	−i	−1	1	k	−k	−j	j
−i	−i	i	1	−1	−k	k	j	−j
j	j	−j	−k	k	−1	1	i	−i
−j	−j	j	k	−k	1	−1	−i	i
k	k	−k	j	−j	−i	i	−1	1
−k	−k	k	−j	j	i	−i	1	−1

Note that the resulting group is non-commutative; for example ij = −ji. A Cayley table, after the 19th century British Mathematician Arthur Cayley, describes the structure of a In Mathematics, commutativity is the ability to change the order of something without changing the end result Q has the unusual property of being Hamiltonian: every subgroup of Q is a normal subgroup, but the group is non-abelian. In Group theory, a Dedekind group is a group G such that 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 In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. Every Hamiltonian group contains a copy of Q.

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. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law The result is a skew field called the quaternions. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible Quaternions, in Mathematics, are a non-commutative extension of Complex numbers They were first described by the Irish Mathematician Note that this is not quite the group algebra on Q (which would be 8-dimensional). In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach 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}.

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

$\langle x,y \mid x^2 = y^2, yxy^{-1} = x^{-1}\rangle$

One may take, for instance, i = x, j = y and k = xy. In Mathematics, one method of defining a group is by a presentation.

The center and the commutator subgroup of Q is the subgroup {±1}. In Abstract algebra, the center of a group G is the set Z ( G) of all elements in G which commute with all the In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup The factor group Q/{±1} is isomorphic to the Klein four-group V. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 The inner automorphism group of Q is isomorphic to Q modulo its center, and is therefore also isomorphic to the Klein four-group. In Abstract algebra, an inner automorphism of a group G is a function f: G &rarr G The full automorphism group of Q is isomorphic to S₄, the symmetric group on four letters. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective In Mathematics, the symmetric group on a set X, denoted by S X or Sym( X) is the group whose underlying The outer automorphism group of Q is then S₄/V which is isomorphic to S₃. In Mathematics, the outer automorphism group of a group G is the quotient of the Automorphism group Aut( G) by its Inner

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). In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

1 Matrix representation of the quaternion group
2 Generalized quaternion group
3 References
4 See also

Matrix representation of the quaternion group

The quaternion group can be represented as a subgroup of the general linear group GL₂(C). In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation

$Q=\{\pm 1, \pm i, \pm j, \pm k\}$ such that

$\ 1= \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$

$\ i= \begin{pmatrix} i & 0 \\ 0 & -i \end{pmatrix}$

$\ j= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$

$\ k= \begin{pmatrix} 0 & i \\ i & 0 \end{pmatrix}$

note: the $i$ s inside the matrices represent the imaginary number $i$ .

The same identities already established in this article can be affirmed using the existing laws of composition for GL₂(C). ^[1]

Generalized quaternion group

A group is called a generalized quaternion group if it has a presentation

$\langle x,y \mid x^{2^{n-1}} = 1, x^{2^{n-2}} = y^2, yxy^{-1} = x^{-1}\rangle$

for some integer n ≥ 3. In Mathematics, one method of defining a group is by a presentation. 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

$x = e^{2\pi i/2^{n-1}}$

$y = j\,$

The generalized quaternion groups are members of the still larger family of dicyclic groups. In Group theory, a dicyclic group is a member of a class of groups Dic n ( n > 1 a Non-abelian group of order 4 n 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 Mathematics, given a Prime number p, a p -group is a Periodic group in which each element has a power of p Another characterization is that a finite p-group in which there is a unique subgroup of order p is either cyclic or generalized quaternion. In particular, for a finite field F with odd characteristic, the 2-Sylow subgroup of SL₂(F) is non-abelian and has only one subgroup of order 2, so this 2-Sylow subgroup must be a generalized quaternion group. Letting q = p^r be the size of F, where p is prime, the size of the 2-Sylow subgroup of SL₂(F) is 2ⁿ, where n = ord₂(p² - 1) + ord₂(r).

References

^ Michael Artin (1991). Michael Artin (born 1934 is an American Mathematician and a professor at MIT, known for his contributions to Algebraic Algebra. Prentice Hall. ISBN 9780130047632.

Contents

Matrix representation of the quaternion group

Generalized quaternion group

References

See also