In
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...
, the quaternion group Q
8 (sometimes just denoted by Q) is a
non-abelian group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
of
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
eight, isomorphic to the eight-element subset
of the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. The algebra of quater ...
s under multiplication. It is given by the
group presentation
In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
:
where ''e'' is the identity element and
commutes with the other elements of the group. These relations, discovered by
W. R. Hamilton, also generate the quaternions as an algebra over the real numbers.
Another presentation of Q
8 is
:
Like many other finite groups, it
can be realized as the
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...
of a certain field of
algebraic number
In mathematics, an algebraic number is a number that is a root of a function, root of a non-zero polynomial in one variable with integer (or, equivalently, Rational number, rational) coefficients. For example, the golden ratio (1 + \sqrt)/2 is ...
s.
Compared to dihedral group
The quaternion group Q
8 has the same order as the
dihedral group
In mathematics, a dihedral group is the group (mathematics), group of symmetry, symmetries of a regular polygon, which includes rotational symmetry, rotations and reflection symmetry, reflections. Dihedral groups are among the simplest example ...
D4, but a different structure, as shown by their Cayley and cycle graphs:
In the diagrams for D
4, the group elements are marked with their action on a letter F in the defining representation R
2. The same cannot be done for Q
8, since it has no faithful representation in R
2 or R
3. D
4 can be realized as a subset of the
split-quaternions in the same way that Q
8 can be viewed as a subset of the quaternions.
Cayley table
The
Cayley table
Named after the 19th-century United Kingdom, British mathematician Arthur Cayley, a Cayley table describes the structure of a finite group by arranging all the possible products of all the group's elements in a square table reminiscent of an additi ...
(multiplication table) for Q
8 is given by:
Properties
The elements ''i'', ''j'', and ''k'' all have
order
Order, ORDER or Orders may refer to:
* A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica
* Categorization, the process in which ideas and objects are recognized, differentiated, and understood
...
four in Q
8 and any two of them generate the entire group. Another
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
of Q
8 based in only two elements to skip this redundancy is:
:
For instance, writing the group elements in
lexicographically minimal normal forms, one may identify:
The quaternion group has the unusual property of being
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
: Q
8 is non-abelian, but every
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation  ...
is
normal. Every Hamiltonian group contains a copy of Q
8.
The quaternion group Q
8 and the dihedral group D
4 are the two smallest examples of a
nilpotent
In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term, along with its sister Idempotent (ring theory), idem ...
non-abelian group.
The
center and the
commutator subgroup
In mathematics, more specifically in abstract algebra, the commutator subgroup or derived subgroup of a group is the subgroup generated by all the commutators of the group.
The commutator subgroup is important because it is the smallest normal ...
of Q
8 is the subgroup
. The
inner automorphism group
In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra given by the conjugation action of a fixed element, called the ''conjugating element''. They can be realized via operations from within the group itself, ...
of Q
8 is given by the group modulo its center, i.e. the
factor group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For exam ...
which is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to the
Klein four-group
In mathematics, the Klein four-group is an abelian group with four elements, in which each element is Involution (mathematics), self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identi ...
V. The full
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
of Q
8 is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to S
4, the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric grou ...
on four letters (see ''Matrix representations'' below), and the
outer automorphism group
In mathematics, the outer automorphism group of a group, , is the quotient, , where is the automorphism group of and ) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted . If is trivial and has ...
of Q
8 is thus S
4/V, which is isomorphic to S
3.
The quaternion group Q
8 has five conjugacy classes,
and so five
irreducible representation
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper nontrivial subrepresentation (\rho, _W, ...
s over the complex numbers, with dimensions 1, 1, 1, 1, 2:
Trivial representation.
Sign representations with i, j, k-kernel: Q
8 has three maximal normal subgroups: the cyclic subgroups generated by i, j, and k respectively. For each maximal normal subgroup ''N'', we obtain a one-dimensional representation factoring through the 2-element
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored out"). For ex ...
''G''/''N''. The representation sends elements of ''N'' to 1, and elements outside ''N'' to −1.
2-dimensional representation: Described below in ''Matrix representations''. It is not
realizable over the real numbers, but is a complex representation: indeed, it is just the quaternions
considered as an algebra over
, and the action is that of left multiplication by
.
The
character table of Q
8 turns out to be the same as that of D
4:
Nevertheless, all the irreducible characters
in the rows above have real values, this gives the
decomposition
Decomposition is the process by which dead organic substances are broken down into simpler organic or inorganic matter such as carbon dioxide, water, simple sugars and mineral salts. The process is a part of the nutrient cycle and is ess ...
of the real
group algebra of
into minimal two-sided
ideals:
:
where the
idempotents