In

groups
A group is a number of people 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 identi ...

); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of

rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery)
A ring is a round band, usually of metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...

that generalize the set of integers. Using tools of algebraic number theory,

A Book of Abstract Algebra

', second edition, from University of Maryland {{DEFAULTSORT:Abstract Algebra Abstract algebra,

algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, which is a broad division of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ...

, abstract algebra (occasionally called modern algebra) is the study of algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. Algebraic structures include groups
A group is a number of people 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 identi ...

, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery)
A ring is a round band, usually of metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...

, fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

, modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a syst ...

, vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, lattices, and algebras
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

. The term ''abstract algebra'' was coined in the early 20th century to distinguish this area of study from older parts of algebra, and more specifically from elementary algebra
Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with spec ...

, the use of variables to represent numbers in computation and reasoning.
Algebraic structures, with their associated homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homom ...

s, form mathematical categories. Category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled directed edges are cal ...

is a formalism that allows a unified way for expressing properties and constructions that are similar for various structures.
Universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular Group (mathematics), groups as t ...

is a related subject that studies types of algebraic structures as single objects. For example, the structure of groups is a single object in universal algebra, which is called the ''variety
Variety may refer to:
Science and technology
Mathematics
* Algebraic variety, the set of solutions of a system of polynomial equations
* Variety (universal algebra), classes of algebraic structures defined by equations in universal algebra
Hort ...

of groups''.
History

As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, manyperhaps mostof these problems were in some way related to the theory ofalgebraic equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

s. Major themes include:
* Solving of systems of linear equations, which led to linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces and through matrix (math ...

* Attempts to find formulas for solutions of general polynomial
In mathematics, a polynomial is an expression (mathematics), expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addition, subtra ...

equations of higher degree that resulted in discovery of groups
A group is a number of people 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 identi ...

as abstract manifestations of symmetry
Symmetry (from Greek συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more pre ...

* Arithmetical investigations of quadratic and higher-degree form
Form is the shape, visual appearance, or :wikt:configuration, configuration of an object. In a wider sense, the form is the way something happens.
Form also refers to:
*Form (document), a document (printed or electronic) with spaces in which to w ...

s and diophantine equation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s, that directly produced the notions of a ring and ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...

.
Numerous textbooks in abstract algebra start with axiomatic
An axiom, postulate or assumption is a statement that is taken to be true
True most commonly refers to truth
Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, ...

definitions of various algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s and then proceed to establish their properties. This creates a false impression that in algebra axioms had come first and then served as a motivation and as a basis of further study. The true order of historical development was almost exactly the opposite. For example, the hypercomplex number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s of the nineteenth century had kinematic
Kinematics is a subfield of physics, developed in classical mechanics, that describes the motion
Image:Leaving Yongsan Station.jpg, 300px, Motion involves a change in position
In physics, motion is the phenomenon in which an object changes its ...

s and physical motivations but challenged comprehension. Most theories that are now recognized as parts of algebra started as collections of disparate facts from various branches of mathematics, acquired a common theme that served as a core around which various results were grouped, and finally became unified on a basis of a common set of concepts. An archetypical example of this progressive synthesis can be seen in the history of group theoryThe history of group theory, a mathematics, mathematical domain studying group (mathematics), groups in their various forms, has evolved in various parallel threads. There are three historical roots of group theory: the theory of algebraic equations, ...

.
Early group theory

There were several threads in the early development of group theory, in modern language loosely corresponding to ''number theory'', ''theory of equations'', and ''geometry''.Leonhard Euler
Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who made important and influential discoveries in many branches of mathematics, such as infinitesimal c ...

considered algebraic operation
In mathematics, a basic algebraic operation is any one of the common Operation (mathematics), operations of arithmetic, which include addition, subtraction, multiplication, Division (mathematics), division, raising to an integer exponentiation, powe ...

s on numbers modulo an integer—modular arithmetic#REDIRECT Modular arithmetic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathemat ...

—in his generalization of Fermat's little theorem
Fermat's little theorem states that if is a prime number, then for any integer , the number is an integer multiple of . In the notation of modular arithmetic, this is expressed as
:a^p \equiv a \pmod p.
For example, if = 2 and = 7, then 27 = ...

. These investigations were taken much further by Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes referr ...

, who considered the structure of multiplicative groups of residues mod n and established many properties of cyclic
Cycle or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in social scienc ...

and more general abelian groups that arise in this way. In his investigations of composition of binary quadratic forms, Gauss explicitly stated the associative law
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

for the composition of forms, but like Euler before him, he seems to have been more interested in concrete results than in general theory. In 1870, Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

gave a definition of an abelian group in the context of ideal class group
In number theory, the ideal class group (or class group) of an algebraic number field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure ( ...

s of a number field, generalizing Gauss's work; but it appears he did not tie his definition with previous work on groups, particularly permutation groups. In 1882, considering the same question, Heinrich M. Weber realized the connection and gave a similar definition that involved the cancellation property
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

but omitted the existence of the inverse element
In abstract algebra, the idea of an inverse element generalises the concepts of negation (sign reversal) (in relation to addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic
Arithmetic ...

, which was sufficient in his context (finite groups).
Permutation
In mathematics, a permutation of a Set (mathematics), set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers ...

s were studied by Joseph-Louis Lagrange
Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaLagrange resolvents
In Galois theory
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analys ...

. Lagrange's goal was to understand why equations of third and fourth degree admit formulas for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the abstract view of the roots, i.e. as symbols and not as numbers. However, he did not consider composition of permutations. Serendipitously, the first edition of Edward Waring
Edward Waring (15 August 1798) was a UK, British mathematician. He entered Magdalene College, Cambridge as a sizar and became Senior wrangler in 1757. He was elected a Fellow of Magdalene and in 1760 Lucasian Professor of Mathematics, holding t ...

's ''Meditationes Algebraicae'' (''Meditations on Algebra'') appeared in the same year, with an expanded version published in 1782. Waring proved the fundamental theorem of symmetric polynomials
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary symm ...

, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. ''Mémoire sur la résolution des équations'' (''Memoire on the Solving of Equations'') of Alexandre Vandermonde (1771) developed the theory of symmetric functions from a slightly different angle, but like Lagrange, with the goal of understanding solvability of algebraic equations.
Paolo Ruffini
Paolo Ruffini (September 22, 1765 – May 10, 1822) was an Italian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

was the first person to develop the theory of permutation group
In mathematics, a permutation group is a group (mathematics), group ''G'' whose elements are permutations of a given Set (mathematics), set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as biject ...

s, and like his predecessors, also in the context of solving algebraic equations. His goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four. En route to this goal he introduced the notion of the order of an element of a group, conjugacy, the cycle decomposition of elements of permutation groups and the notions of primitive and imprimitive and proved some important theorems relating these concepts, such as
However, he got by without formalizing the concept of a group, or even of a permutation group.
The next step was taken by Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by Nth root, radical ...

in 1832, although his work remained unpublished until 1846, when he considered for the first time what is now called the ''closure property'' of a group of permutations, which he expressed as
The theory of permutation groups received further far-reaching development in the hands of Augustin Cauchy
Baron
Baron is a rank of nobility or title of honour, often hereditary, in various European countries, either current or historical. The female equivalent is baroness. Typically, the title denotes an aristocrat who ranks higher than a lord ...

and Camille Jordan
Marie Ennemond Camille Jordan (; 5 January 1838 – 22 January 1922) was a French mathematician, known both for his foundational work in group theory and for his influential ''Cours d'analyse''.
Biography
Jordan was born in Lyon and educated at ...

, both through introduction of new concepts and, primarily, a great wealth of results about special classes of permutation groups and even some general theorems. Among other things, Jordan defined a notion of isomorphism
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

, still in the context of permutation groups and, incidentally, it was he who put the term ''group'' in wide use.
The abstract notion of a group appeared for the first time in Arthur Cayley
Arthur Cayley (; 16 August 1821 – 26 January 1895) was a prolific British mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such ...

's papers in 1854. Cayley realized that a group need not be a permutation group (or even ''finite''), and may instead consist of matrices
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics)
In mathematics, a matrix (plural matrices) is a rectangle, rectangular ''wikt:array, array'' or ''table'' of numbers, symbol (formal), symbols, or expression (mathema ...

, whose algebraic properties, such as multiplication and inverses, he systematically investigated in succeeding years. Much later Cayley would revisit the question whether abstract groups were more general than permutation groups, and establish that, in fact, any group is isomorphic to a group of permutations.
Modern algebra

The end of the 19th and the beginning of the 20th century saw a shift in the methodology of mathematics. Abstract algebra emerged around the start of the 20th century, under the name ''modern algebra''. Its study was part of the drive for more intellectual rigor in mathematics. Initially, the assumptions in classicalalgebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, on which the whole of mathematics (and major parts of the natural sciences
Natural science is a Branches of science, branch of science concerned with the description, understanding and prediction of Phenomenon, natural phenomena, based on empirical evidence from observation and experimentation. Mechanisms such as peer r ...

) depend, took the form of axiomatic system
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s. No longer satisfied with establishing properties of concrete objects, mathematicians started to turn their attention to general theory. Formal definitions of certain algebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s began to emerge in the 19th century. For example, results about various groups of permutations came to be seen as instances of general theorems that concern a general notion of an ''abstract group''. Questions of structure and classification of various mathematical objects came to forefront.
These processes were occurring throughout all of mathematics, but became especially pronounced in algebra. Formal definition through primitive operations and axioms were proposed for many basic algebraic structures, such as groups
A group is a number of people 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 identi ...

, rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery)
A ring is a round band, usually of metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...

, and fields
File:A NASA Delta IV Heavy rocket launches the Parker Solar Probe (29097299447).jpg, FIELDS heads into space in August 2018 as part of the ''Parker Solar Probe''
FIELDS is a science instrument on the ''Parker Solar Probe'' (PSP), designed to mea ...

. Hence such things as group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...

and ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure ...

took their places in pure mathematics
Pure mathematics is the study of mathematical concepts independently of any application outside mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, struc ...

. The algebraic investigations of general fields by Ernst Steinitz
Ernst Steinitz (13 June 1871 – 29 September 1928) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quant ...

and of commutative and then general rings by David Hilbert
David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician and one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in man ...

, Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as qua ...

and Emmy Noether
Amalie Emmy Noether Emmy is the '' Rufname'', the second of two official given names, intended for daily use. Cf. for example the résumé submitted by Noether to Erlangen University in 1907 (Erlangen University archive, ''Promotionsakt Emmy Noeth ...

, building up on the work of Ernst Kummer
Ernst Eduard Kummer (29 January 1810 – 14 May 1893) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quan ...

, Leopold Kronecker
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics a ...

and Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to abstract algebra (particularly ring theory
In algebra, ring theory is the study of ring (mathematics), rings ...

, who had considered ideals in commutative rings, and of Georg Frobenius
Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German
German(s) may refer to:
Common uses
* of or related to Germany
* Germans, Germanic ethnic group, citizens of Germany or people of German ancestry
* For citizens of Germa ...

and Issai Schur
Issai Schur (January 10, 1875 – January 10, 1941) was a Russia
Russia (russian: link=no, Россия, , ), or the Russian Federation, is a country spanning Eastern Europe and Northern Asia. It is the List of countries and dependencies b ...

, concerning representation theory
Representation theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change ( ...

of groups, came to define abstract algebra. These developments of the last quarter of the 19th century and the first quarter of 20th century were systematically exposed in Bartel van der Waerden's ''Moderne Algebra
''Moderne Algebra'' is a two-volume German textbook on graduate abstract algebra by , originally based on lectures given by Emil Artin in 1926 and by from 1924 to 1928. The English translation of 1949–1950 had the title ''Modern algebra'', thoug ...

'', the two-volume monograph
A monograph is a specialist work of writing (in contrast to reference work
A reference work is a work such as a book
A book is a medium for recording information
Information can be thought of as the resolution of uncertainty; it a ...

published in 1930–1931 that forever changed for the mathematical world the meaning of the word ''algebra'' from ''the theory of equations'' to the ''theory of algebraic structures''.
Basic concepts

By abstracting away various amounts of detail, mathematicians have defined various algebraic structures that are used in many areas of mathematics. For instance, almost all systems studied are sets, to which the theorems ofset theory
illustrating the intersection of two sets
Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as ...

apply. Those sets that have a certain binary operation defined on them form magmas
300px, Hawaii
Hawaii ( ; haw, Hawaii or ) is a U.S. state in the Western United States, in the Pacific Ocean about 2,000 miles (3,200 km) from the U.S. mainland. It is the only state outside North America, the only island state, and ...

, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form semigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...

s); identity, and inverses (to form group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Ernő Rubik has bee ...

may be used when studying rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery)
A ring is a round band, usually of metal
A metal (from Ancient Greek, Greek μέταλλον ''métallon'', "mine ...

(algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. In general there is a balance between the amount of generality and the richness of the theory: more general structures have usually fewer nontrivial
In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., group (mathematics), groups, topological spaces). The noun trivialit ...

theorems and fewer applications.
Examples of algebraic structures with a single binary operation
In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an Operation (mathematics), operation of arity two.
More specif ...

are:
* Magma
300px, Hawaii
Hawaii ( ; haw, Hawaii or ) is a U.S. state, state in the Western United States, located in the Pacific Ocean about 2,000 miles from the U.S. mainland. It is the only state outside North America, the only state that is ...

* Quasigroup
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

* Monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematic ...

* Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative binary operation.
The binary operation of a semigroup is most often denoted multiplication, multiplicatively: ''x''·''y'', o ...

* Group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

Examples involving several operations include:
* Ring
* Field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

* Module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modula ...

* Vector space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

* Algebra over a field
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

* Associative algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

* Lie algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

* Lattice
* Boolean algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

Applications

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance,algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...

uses algebraic objects to study topologies. The Poincaré conjecture
In mathematics, the Poincaré conjecture (, , ) is a theorem about the Characterization (mathematics), characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space.
The conjecture states: An equ ...

, proved in 2003, asserts that the fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence class
In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory
Title page of the first edition of Disquisitiones Arithmeticae, one of the founding works of modern algebraic number theory.
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the intege ...

studies various number Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specializing in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Fermat ...

proved Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than 2. The cases ...

.
In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory
In physics, a gauge theory is a type of Field theory (physics), field theory in which the Lagrangian (field theory), Lagrangian (and hence the dynamics of the system itself) does not change (is Invariant (physics), invariant) under local symmetry, ...

, the requirement of local symmetry
In physics, a symmetry of a physical system
In physics
Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its ...

can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to the dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian.
See also

* Coding theory * Group theory * List of publications in abstract algebraReferences

Sources

* * * * * * * * W. Keith Nicholson (2012) ''Introduction to Abstract Algebra'', 4th edition, John Wiley & Sons . * John R. Durbin (1992) ''Modern Algebra : an introduction'', John Wiley & SonsExternal links

* Charles C. Pinter (1990) [1982]A Book of Abstract Algebra

', second edition, from University of Maryland {{DEFAULTSORT:Abstract Algebra Abstract algebra,