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# Quaternion group

In group theory, the quaternion group Q8 (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication. [1]

58 relations: Abelian group, Abstract algebra, American Mathematical Monthly, Automorphism, Binary tetrahedral group, Biquaternion, Brauer–Suzuki theorem, Cambridge University Press, Cayley graph, Cayley table, Center (group theory), Clifford algebra, Commutative property, Commutator subgroup, Cycle graph (algebra), Dedekind group, Dicyclic group, Dihedral group, Division ring, European Journal of Physics, Examples of groups, Finite field, Fundamental theorem of Galois theory, Galois group, General linear group, Group (mathematics), Group algebra, Group representation, Group ring, Group theory, Harold Scott MacDonald Coxeter, Hurwitz quaternion, Ideal (ring theory), Index of a subgroup, Inner automorphism, Isomorphism, Klein four-group, List of small groups, Non-abelian group, Normal subgroup, Order (group theory), Outer automorphism group, P-group, Point groups in three dimensions, Polyhedral group, Presentation of a group, Quaternion, Quotient group, Quotient ring, Rational number, ... Expand index (8 more) »

## Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written.

## Abstract algebra

In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures.

## American Mathematical Monthly

The American Mathematical Monthly is a mathematical journal founded by Benjamin Finkel in 1894.

## Automorphism

In mathematics, an automorphism is an isomorphism from a mathematical object to itself.

## Binary tetrahedral group

In mathematics, the binary tetrahedral group, denoted 2T or 2,3,3 is a certain nonabelian group of order 24.

## Biquaternion

In abstract algebra, the biquaternions are the numbers, where, and are complex numbers, or variants thereof, and the elements of multiply as in the quaternion group.

## Brauer–Suzuki theorem

In mathematics, the Brauer–Suzuki theorem, proved by,,, states that if a finite group has a generalized quaternion Sylow 2-subgroup and no non-trivial normal subgroups of odd order, then the group has a centre of order 2.

## Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

## Cayley graph

In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group.

## Cayley table

A Cayley table, after the 19th century British mathematician Arthur Cayley, 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 addition or multiplication table.

## Center (group theory)

In abstract algebra, the center of a group,, is the set of elements that commute with every element of.

## Clifford algebra

In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra.

## Commutative property

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result.

## 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.

## Cycle graph (algebra)

In group theory, a sub-field of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.

## Dedekind group

In group theory, a Dedekind group is a group G such that every subgroup of G is normal.

## Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n) is a member of a class of non-abelian groups of order 4n (n > 1).

## Dihedral group

In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections.

## Division ring

In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible.

## European Journal of Physics

The European Journal of Physics is a peer-reviewed, scientific journal dedicated to maintaining and improving the standard of physics education in higher education.

## Examples of groups

Some elementary examples of groups in mathematics are given on Group (mathematics).

## Finite field

In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.

## Fundamental theorem of Galois theory

In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions.

## Galois group

In mathematics, more specifically 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.

## General linear group

In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.

## Group (mathematics)

In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.

## Group algebra

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 algebra), such that representations of the algebra are related to representations of the group.

## Group representation

In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.

## Group ring

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group.

## Group theory

In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.

## Harold Scott MacDonald Coxeter

Harold Scott MacDonald "Donald" Coxeter, FRS, FRSC, (February 9, 1907 &ndash; March 31, 2003) was a British-born Canadian geometer.

## Hurwitz quaternion

In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded).

## Ideal (ring theory)

In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring.

## Index of a subgroup

In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).

## Inner automorphism

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.

## Isomorphism

In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.

## Klein four-group

In mathematics, the Klein four-group (or just Klein group or Vierergruppe, four-group, often symbolized by the letter V or as K4) is the group, the direct product of two copies of the cyclic group of order 2.

## List of small groups

The following list in mathematics contains the finite groups of small order up to group isomorphism.

## Non-abelian group

In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a.

## Normal subgroup

In abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group of which it is a part.

## Order (group theory)

In group theory, a branch of mathematics, the term order is used in two unrelated senses.

## 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.

## P-group

In mathematical group theory, given a prime number p, a p-group is a group in which each element has a power of p as its order.

## Point groups in three dimensions

In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere.

## Polyhedral group

In geometry, the polyhedral group is any of the symmetry groups of the Platonic solids.

## Presentation of a group

In mathematics, one method of defining a group is by a presentation.

## Quaternion

In mathematics, the quaternions are a number system that extends the complex numbers.

## 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 the group structure.

## Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient groups of group theory and the quotient spaces of linear algebra.

## Rational number

In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.

## Special linear group

In mathematics, the special linear group of degree n over a field F is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion.

## Split-quaternion

In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name.

## Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial splits or decomposes into linear factors.

## Subgroup

In group theory, a branch of mathematics, given a group G under a binary operation ∗, a subset H of G is called a subgroup of G if H also forms a group under the operation ∗.

## 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.

## Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

## Wolfram Alpha

Wolfram Alpha (also styled WolframAlpha, and Wolfram|Alpha) is a computational knowledge engine or answer engine developed by Wolfram Alpha LLC, a subsidiary of Wolfram Research.

## 16-cell

In four-dimensional geometry, a 16-cell is a regular convex 4-polytope.

## References

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