Inmathematics, agroup is aset with abinary operation that satisfies the following constraints: the operation isassociative, it has anidentity element, and every element of the set has aninverse element.

Manymathematical structures are groups endowed with other properties. For example, theintegers with theaddition operation form aninfinite group, which isgenerated by a single element called⁠${\displaystyle 1}$⁠ (these properties characterize the integers in a unique way).

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers,geometric shapes andpolynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics.^[1][2]

Ingeometry, groups arise naturally in the study ofsymmetries andgeometric transformations: The symmetries of an object form a group, called thesymmetry group of the object, and the transformations of a given type form a general group.Lie groups appear in symmetry groups in geometry, and also in theStandard Model ofparticle physics. ThePoincaré group is a Lie group consisting of the symmetries ofspacetime inspecial relativity.Point groups describesymmetry in molecular chemistry.

The concept of a group arose in the study ofpolynomial equations, starting withÉvariste Galois in the 1830s, who introduced the termgroup (French:groupe) for the symmetry group of theroots of an equation, now called aGalois group. After contributions from other fields such asnumber theory and geometry, the group notion was generalized and firmly established around 1870. Moderngroup theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such assubgroups,quotient groups andsimple groups. In addition to their abstract properties, group theorists also study the different ways in which a group can be expressed concretely, both from a point of view ofrepresentation theory (that is, through therepresentations of the group) and ofcomputational group theory. A theory has been developed forfinite groups, which culminated with theclassification of finite simple groups, completed in 2004. Since the mid-1980s,geometric group theory, which studiesfinitely generated groups as geometric objects, has become an active area in group theory.

One of the more familiar groups is the set ofintegers$${\displaystyle \mathbb{Z}=\{\dots ,-4,-3,-2,-1,0,1,2,3,4,\dots \}}$$ together withaddition.^[3] For any two integers${\displaystyle a}$  and⁠${\displaystyle b}$ ⁠, thesum${\displaystyle a+b}$  is also an integer; thisclosure property says that${\displaystyle +}$  is abinary operation on⁠${\displaystyle \mathbb{Z}}$ ⁠. The following properties of integer addition serve as a model for the group axioms in the definition below.

• For all integers⁠${\displaystyle a}$ ⁠,${\displaystyle b}$  and ⁠${\displaystyle c}$ ⁠, one has⁠${\displaystyle (a+b)+c=a+(b+c)}$ ⁠. Expressed in words, adding${\displaystyle a}$  to${\displaystyle b}$  first, and then adding the result to${\displaystyle c}$  gives the same final result as adding${\displaystyle a}$  to the sum of${\displaystyle b}$  and ⁠${\displaystyle c}$ ⁠. This property is known asassociativity.
• If${\displaystyle a}$  is any integer, then${\displaystyle 0+a=a}$  and⁠${\displaystyle a+0=a}$ ⁠.Zero is called theidentity element of addition because adding it to any integer returns the same integer.
• For every integer⁠${\displaystyle a}$ ⁠, there is an integer${\displaystyle b}$  such that${\displaystyle a+b=0}$  and⁠${\displaystyle b+a=0}$ ⁠. The integer${\displaystyle b}$  is called theinverse element of the integer${\displaystyle a}$  and is denoted ⁠${\displaystyle -a}$ ⁠.

The integers, together with the operation⁠${\displaystyle +}$ ⁠, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.

A group is a non-emptyset${\displaystyle G}$  together with abinary operation on⁠${\displaystyle G}$ ⁠, here denoted "⁠${\displaystyle \cdot }$ ⁠", that combines any twoelements${\displaystyle a}$  and${\displaystyle b}$  of${\displaystyle G}$  to form an element of⁠${\displaystyle G}$ ⁠, denoted⁠${\displaystyle a\cdot b}$ ⁠, such that the following three requirements, known asgroup axioms, are satisfied:^[5][6][7][a]

Associativity

For all⁠${\displaystyle a}$ ⁠,⁠${\displaystyle b}$ ⁠,⁠${\displaystyle c}$ ⁠ in⁠${\displaystyle G}$ ⁠, one has⁠${\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)}$ ⁠.

Identity element

There exists an element${\displaystyle e}$  in${\displaystyle G}$  such that, for every${\displaystyle a}$  in⁠${\displaystyle G}$ ⁠, one has⁠${\displaystyle e\cdot a=a}$ ⁠ and⁠${\displaystyle a\cdot e=a}$ ⁠.

Such an element is unique (see below). It is called theidentity element (or sometimesneutral element) of the group.

Inverse element

For each${\displaystyle a}$  in⁠${\displaystyle G}$ ⁠, there exists an element${\displaystyle b}$  in${\displaystyle G}$  such that${\displaystyle a\cdot b=e}$  and⁠${\displaystyle b\cdot a=e}$ ⁠, where${\displaystyle e}$  is the identity element.

For each⁠${\displaystyle a}$ ⁠, the element${\displaystyle b}$  is unique (see below); it is calledthe inverse of${\displaystyle a}$  and is commonly denoted⁠${\displaystyle a^{-1}}$ ⁠.

Formally, a group is anordered pair of a set and a binary operation on this set that satisfies thegroup axioms. The set is called theunderlying set of the group, and the operation is called thegroup operation or thegroup law.

A group and its underlying set are thus two differentmathematical objects. To avoid cumbersome notation, it is common toabuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.

For example, consider the set ofreal numbers⁠${\displaystyle \mathbb{R}}$ ⁠, which has the operations of addition${\displaystyle a+b}$  andmultiplication⁠${\displaystyle ab}$ ⁠. Formally,${\displaystyle \mathbb{R}}$  is a set,${\displaystyle (\mathbb{R},+)}$  is a group, and${\displaystyle (\mathbb{R},+,\cdot )}$  is afield. But it is common to write${\displaystyle \mathbb{R}}$  to denote any of these three objects.

Theadditive group of the field${\displaystyle \mathbb{R}}$  is the group whose underlying set is${\displaystyle \mathbb{R}}$  and whose operation is addition. Themultiplicative group of the field${\displaystyle \mathbb{R}}$  is the group${\displaystyle {\mathbb{R}}^{\times }}$  whose underlying set is the set of nonzero real numbers${\displaystyle \mathbb{R}\setminus \{0\}}$  and whose operation is multiplication.

More generally, one speaks of anadditive group whenever the group operation is notated as addition; in this case, the identity is typically denoted⁠${\displaystyle 0}$ ⁠, and the inverse of an element${\displaystyle x}$  is denoted⁠${\displaystyle -x}$ ⁠. Similarly, one speaks of amultiplicative group whenever the group operation is notated as multiplication; in this case, the identity is typically denoted⁠${\displaystyle 1}$ ⁠, and the inverse of an element${\displaystyle x}$  is denoted⁠${\displaystyle x^{-1}}$ ⁠. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition,${\displaystyle ab}$  instead of⁠${\displaystyle a\cdot b}$ ⁠.

The definition of a group does not require that${\displaystyle a\cdot b=b\cdot a}$  for all elements${\displaystyle a}$  and${\displaystyle b}$  in⁠${\displaystyle G}$ ⁠. If this additional condition holds, then the operation is said to becommutative, and the group is called anabelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.

Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements arefunctions, the operation is oftenfunction composition⁠${\displaystyle f\circ g}$ ⁠; then the identity may be denoted id. In the more specific cases ofgeometric transformation groups,symmetry groups,permutation groups, andautomorphism groups, the symbol${\displaystyle \circ }$  is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

Two figures in theplane arecongruent if one can be changed into the other using a combination ofrotations,reflections, andtranslations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are calledsymmetries. Asquare has eight symmetries. These are:

• theidentity operation leaving everything unchanged, denoted id;
• rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by⁠${\displaystyle r_1}$ ⁠,${\displaystyle r_2}$  and⁠${\displaystyle r_3}$ ⁠, respectively;
• reflections about the horizontal and vertical middle line (⁠${\displaystyle f_{\mathrm{v}}}$ ⁠ and⁠${\displaystyle f_{\mathrm{h}}}$ ⁠), or through the twodiagonals (⁠${\displaystyle f_{\mathrm{d}}}$ ⁠ and⁠${\displaystyle f_{\mathrm{c}}}$ ⁠).

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example,${\displaystyle r_1}$  sends a point to its rotation 90° clockwise around the square's center, and${\displaystyle f_{\mathrm{h}}}$  sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called thedihedral group of degree four, denoted⁠${\displaystyle {\mathrm{D}}_4}$ ⁠. The underlying set of the group is the above set of symmetries, and the group operation is function composition.^[8] Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first${\displaystyle a}$  and then${\displaystyle b}$  is written symbolicallyfrom right to left as${\displaystyle b\circ a}$  ("apply the symmetry${\displaystyle b}$  after performing the symmetry⁠${\displaystyle a}$ ⁠"). This is the usual notation for composition of functions.

ACayley table lists the results of all such compositions possible. For example, rotating by 270° clockwise (⁠${\displaystyle r_3}$ ⁠) and then reflecting horizontally (⁠${\displaystyle f_{\mathrm{h}}}$ ⁠) is the same as performing a reflection along the diagonal (⁠${\displaystyle f_{\mathrm{d}}}$ ⁠). Using the above symbols, highlighted in blue in the Cayley table:$${\displaystyle f_{\mathrm{h}}\circ r_3=f_{\mathrm{d}}.}$$ 

Given this set of symmetries and the described operation, the group axioms can be understood as follows.

Binary operation: Composition is a binary operation. That is,${\displaystyle a\circ b}$  is a symmetry for any two symmetries${\displaystyle a}$  and⁠${\displaystyle b}$ ⁠. For example,$${\displaystyle r_3\circ f_{\mathrm{h}}=f_{\mathrm{c}},}$$ that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (⁠${\displaystyle f_{\mathrm{c}}}$ ⁠). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the Cayley table.

Associativity: The associativity axiom deals with composing more than two symmetries: Starting with three elements⁠${\displaystyle a}$ ⁠,⁠${\displaystyle b}$ ⁠ and⁠${\displaystyle c}$ ⁠ of⁠${\displaystyle {\mathrm{D}}_4}$ ⁠, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose${\displaystyle a}$  and${\displaystyle b}$  into a single symmetry, then to compose that symmetry with⁠${\displaystyle c}$ ⁠. The other way is to first compose${\displaystyle b}$  and⁠${\displaystyle c}$ ⁠, then to compose the resulting symmetry with⁠${\displaystyle a}$ ⁠. These two ways must give always the same result, that is,$${\displaystyle (a\circ b)\circ c=a\circ (b\circ c),}$$ For example,${\displaystyle (f_{\mathrm{d}}\circ f_{\mathrm{v}})\circ r_2=f_{\mathrm{d}}\circ (f_{\mathrm{v}}\circ r_2)}$  can be checked using the Cayley table: 

Identity element: The identity element is⁠${\displaystyle \mathrm{i}\mathrm{d}}$ ⁠, as it does not change any symmetry${\displaystyle a}$  when composed with it either on the left or on the right.

Inverse element: Each symmetry has an inverse:⁠${\displaystyle \mathrm{i}\mathrm{d}}$ ⁠, the reflections⁠${\displaystyle f_{\mathrm{h}}}$ ⁠,⁠${\displaystyle f_{\mathrm{v}}}$ ⁠,⁠${\displaystyle f_{\mathrm{d}}}$ ⁠,⁠${\displaystyle f_{\mathrm{c}}}$ ⁠ and the 180° rotation${\displaystyle r_2}$  are their own inverse, because performing them twice brings the square back to its original orientation. The rotations${\displaystyle r_3}$  and${\displaystyle r_1}$  are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.

In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in⁠${\displaystyle {\mathrm{D}}_4}$ ⁠, as, for example,${\displaystyle f_{\mathrm{h}}\circ r_1=f_{\mathrm{c}}}$  but⁠${\displaystyle r_1\circ f_{\mathrm{h}}=f_{\mathrm{d}}}$ ⁠. In other words,${\displaystyle {\mathrm{D}}_4}$  is not abelian.

The modern concept of anabstract group developed out of several fields of mathematics.^[9][10][11] The original motivation for group theory was the quest for solutions ofpolynomial equations of degree higher than 4. The 19th-century French mathematicianÉvariste Galois, extending prior work ofPaolo Ruffini andJoseph-Louis Lagrange, gave a criterion for thesolvability of a particular polynomial equation in terms of thesymmetry group of itsroots (solutions). The elements of such aGalois group correspond to certainpermutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously.^[12][13] More general permutation groups were investigated in particular byAugustin Louis Cauchy.Arthur Cayley'sOn the theory of groups, as depending on the symbolic equation${\displaystyle {\theta }^n=1}$  (1854) gives the first abstract definition of afinite group.^[14]

Geometry was a second field in which groups were used systematically, especially symmetry groups as part ofFelix Klein's 1872Erlangen program.^[15] After novel geometries such ashyperbolic andprojective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas,Sophus Lie founded the study ofLie groups in 1884.^[16]

The third field contributing to group theory wasnumber theory. Certain abelian group structures had been used implicitly inCarl Friedrich Gauss's number-theoretical workDisquisitiones Arithmeticae (1798), and more explicitly byLeopold Kronecker.^[17] In 1847,Ernst Kummer made early attempts to proveFermat's Last Theorem by developinggroups describing factorization intoprime numbers.^[18]

The convergence of these various sources into a uniform theory of groups started withCamille Jordan'sTraité des substitutions et des équations algébriques (1870).^[19]Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time.^[20] As of the 20th century, groups gained wide recognition by the pioneering work ofFerdinand Georg Frobenius andWilliam Burnside (who worked onrepresentation theory of finite groups),Richard Brauer'smodular representation theory andIssai Schur's papers.^[21] The theory of Lie groups, and more generallylocally compact groups was studied byHermann Weyl,Élie Cartan and many others.^[22] Itsalgebraic counterpart, the theory ofalgebraic groups, was first shaped byClaude Chevalley (from the late 1930s) and later by the work ofArmand Borel andJacques Tits.^[23]

TheUniversity of Chicago's 1960–61 Group Theory Year brought together group theorists such asDaniel Gorenstein,John G. Thompson andWalter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to theclassification of finite simple groups, with the final step taken byAschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length ofproof and number of researchers. Research concerning this classification proof is ongoing.^[24] Group theory remains a highly active mathematical branch,^[b] impacting many other fields, as theexamples below illustrate.

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed underelementary group theory.^[25] For example,repeated applications of the associativity axiom show that the unambiguity of$${\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}$$ generalizes to more than three factors. Because this implies thatparentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.^[26]

The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements${\displaystyle e}$  and${\displaystyle f}$  of a group are equal, because the group axioms imply⁠${\displaystyle e=e\cdot f=f}$ ⁠. It is thus customary to speak ofthe identity element of the group.^[27]

The group axioms also imply that the inverse of each element is unique. Let a group element${\displaystyle a}$  have both${\displaystyle b}$  and${\displaystyle c}$  as inverses. Then

 

Therefore, it is customary to speak ofthe inverse of an element.^[27]

Given elements${\displaystyle a}$  and${\displaystyle b}$  of a group⁠${\displaystyle G}$ ⁠, there is a unique solution${\displaystyle x}$  in${\displaystyle G}$  to the equation⁠${\displaystyle a\cdot x=b}$ ⁠, namely⁠${\displaystyle a^{-1}\cdot b}$ ⁠.^[c][28] It follows that for each${\displaystyle a}$  in${\displaystyle G}$ , the function${\displaystyle G\to G}$  that maps each${\displaystyle x}$  to${\displaystyle a\cdot x}$  is abijection; it is calledleft multiplication by${\displaystyle a}$  orleft translation by⁠${\displaystyle a}$ ⁠.

Similarly, given${\displaystyle a}$  and⁠${\displaystyle b}$ ⁠, the unique solution to${\displaystyle x\cdot a=b}$  is⁠${\displaystyle b\cdot a^{-1}}$ ⁠. For each⁠${\displaystyle a}$ ⁠, the function${\displaystyle G\to G}$  that maps each${\displaystyle x}$  to${\displaystyle x\cdot a}$  is a bijection calledright multiplication by${\displaystyle a}$  orright translation by⁠${\displaystyle a}$ ⁠.

The group axioms for identity and inverses may be "weakened" to assert only the existence of aleft identity andleft inverses. From theseone-sided axioms, one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are not weaker.^[29]

In particular, assuming associativity and the existence of a left identity${\displaystyle e}$  (that is,⁠${\displaystyle e\cdot f=f}$ ⁠) and a left inverse${\displaystyle f^{-1}}$  for each element${\displaystyle f}$  (that is,⁠${\displaystyle f^{-1}\cdot f=e}$ ⁠), one can show that every left inverse is also a right inverse of the same element as follows.^[29]Indeed, one has

 

Similarly, the left identity is also a right identity:^[29]

 

These proofs require all three axioms (associativity, existence of left identity and existence of left inverse). For a structure with a looser definition (like asemigroup) one may have, for example, that a left identity is not necessarily a right identity.

The same result can be obtained by only assuming the existence of a right identity and a right inverse.

However, only assuming the existence of aleft identity and aright inverse (or vice versa) is not sufficient to define a group. For example, consider the set${\displaystyle G=\{e,f\}}$  with the operator${\displaystyle \cdot }$  satisfying${\displaystyle e\cdot e=f\cdot e=e}$  and⁠${\displaystyle e\cdot f=f\cdot f=f}$ ⁠. This structure does have a left identity (namely,⁠${\displaystyle e}$ ⁠), and each element has a right inverse (which is${\displaystyle e}$  for both elements). Furthermore, this operation is associative (since the product of any number of elements is always equal to the rightmost element in that product, regardless of the order in which these operations are done). However,${\displaystyle (G,\cdot )}$  is not a group, since it lacks a right identity.

When studying sets, one uses concepts such assubset, function, andquotient by an equivalence relation. When studying groups, one uses insteadsubgroups,homomorphisms, andquotient groups. These are the analogues that take the group structure into account.^[d]

Group homomorphisms^[e] are functions that respect group structure; they may be used to relate two groups. Ahomomorphism from a group${\displaystyle (G,\cdot )}$  to a group${\displaystyle (H,\ast )}$  is a function${\displaystyle \phi :G\to H}$  such that

It would be natural to require also that${\displaystyle \phi }$  respect identities,⁠${\displaystyle \phi (1_G)=1_H}$ ⁠, and inverses,${\displaystyle \phi (a^{-1})=\phi (a{)}^{-1}}$  for all${\displaystyle a}$  in⁠${\displaystyle G}$ ⁠. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.^[30]

Theidentity homomorphism of a group${\displaystyle G}$  is the homomorphism${\displaystyle {\iota }_G:G\to G}$  that maps each element of${\displaystyle G}$  to itself. Aninverse homomorphism of a homomorphism${\displaystyle \phi :G\to H}$  is a homomorphism${\displaystyle \psi :H\to G}$  such that${\displaystyle \psi \circ \phi ={\iota }_G}$  and⁠${\displaystyle \phi \circ \psi ={\iota }_H}$ ⁠, that is, such that${\displaystyle \psi (\phi (g))=g}$  for all${\displaystyle g}$  in${\displaystyle G}$  and such that${\displaystyle \phi (\psi (h))=h}$  for all${\displaystyle h}$  in⁠${\displaystyle H}$ ⁠. Anisomorphism is a homomorphism that has an inverse homomorphism; equivalently, it is abijective homomorphism. Groups${\displaystyle G}$  and${\displaystyle H}$  are calledisomorphic if there exists an isomorphism⁠${\displaystyle \phi :G\to H}$ ⁠. In this case,${\displaystyle H}$  can be obtained from${\displaystyle G}$  simply by renaming its elements according to the function⁠${\displaystyle \phi }$ ⁠; then any statement true for${\displaystyle G}$  is true for⁠${\displaystyle H}$ ⁠, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form acategory, thecategory of groups.^[31]

Aninjective homomorphism${\displaystyle \varphi :G^{\prime }\to G}$  factors canonically as an isomorphism followed by an inclusion,${\displaystyle G^{\prime }\stackrel{\sim }{\to }H\hookrightarrow G}$  for some subgroup⁠${\displaystyle H}$ ⁠ of⁠${\displaystyle G}$ ⁠.Injective homomorphisms are themonomorphisms in the category of groups.

Informally, asubgroup is a group${\displaystyle H}$  contained within a bigger one,⁠${\displaystyle G}$ ⁠: it has a subset of the elements of⁠${\displaystyle G}$ ⁠, with the same operation.^[32] Concretely, this means that the identity element of${\displaystyle G}$  must be contained in⁠${\displaystyle H}$ ⁠, and whenever${\displaystyle h_1}$  and${\displaystyle h_2}$  are both in⁠${\displaystyle H}$ ⁠, then so are${\displaystyle h_1\cdot h_2}$  and⁠${\displaystyle h_1^{-1}}$ ⁠, so the elements of⁠${\displaystyle H}$ ⁠, equipped with the group operation on${\displaystyle G}$  restricted to⁠${\displaystyle H}$ ⁠, indeed form a group. In this case, the inclusion map${\displaystyle H\to G}$  is a homomorphism.

In the example of symmetries of a square, the identity and the rotations constitute a subgroup⁠${\displaystyle R=\{\mathrm{i}\mathrm{d},r_1,r_2,r_3\}}$ ⁠, highlighted in red in the Cayley table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. Thesubgroup test provides anecessary and sufficient condition for a nonempty subset⁠${\displaystyle H}$ ⁠ of a group⁠${\displaystyle G}$ ⁠ to be a subgroup: it is sufficient to check that${\displaystyle g^{-1}\cdot h\in H}$  for all elements${\displaystyle g}$  and${\displaystyle h}$  in⁠${\displaystyle H}$ ⁠. Knowing a group'ssubgroups is important in understanding the group as a whole.^[f]

Given any subset${\displaystyle S}$  of a group⁠${\displaystyle G}$ ⁠, the subgroupgenerated by${\displaystyle S}$  consists of all products of elements of${\displaystyle S}$  and their inverses. It is the smallest subgroup of${\displaystyle G}$  containing⁠${\displaystyle S}$ ⁠.^[33] In the example of symmetries of a square, the subgroup generated by${\displaystyle r_2}$  and${\displaystyle f_{\mathrm{v}}}$  consists of these two elements, the identity element⁠${\displaystyle \mathrm{i}\mathrm{d}}$ ⁠, and the element⁠${\displaystyle f_{\mathrm{h}}=f_{\mathrm{v}}\cdot r_2}$ ⁠. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.

In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup${\displaystyle H}$  determines left and right cosets, which can be thought of as translations of${\displaystyle H}$  by an arbitrary group element⁠${\displaystyle g}$ ⁠. In symbolic terms, theleft andright cosets of⁠${\displaystyle H}$ ⁠, containing an element⁠${\displaystyle g}$ ⁠, are

The left cosets of any subgroup${\displaystyle H}$  form apartition of⁠${\displaystyle G}$ ⁠; that is, theunion of all left cosets is equal to${\displaystyle G}$  and two left cosets are either equal or have anemptyintersection.^[35] The first case${\displaystyle g_{1}H=g_{2}H}$  happensprecisely when⁠${\displaystyle g_1^{-1}\cdot g_2\in H}$ ⁠, i.e., when the two elements differ by an element of⁠${\displaystyle H}$ ⁠. Similar considerations apply to the right cosets of⁠${\displaystyle H}$ ⁠. The left cosets of${\displaystyle H}$  may or may not be the same as its right cosets. If they are (that is, if all${\displaystyle g}$  in${\displaystyle G}$  satisfy⁠${\displaystyle gH=Hg}$ ⁠), then${\displaystyle H}$  is said to be anormal subgroup.

In⁠${\displaystyle {\mathrm{D}}_4}$ ⁠, the group of symmetries of a square, with its subgroup${\displaystyle R}$  of rotations, the left cosets${\displaystyle gR}$  are either equal to⁠${\displaystyle R}$ ⁠, if${\displaystyle g}$  is an element of${\displaystyle R}$  itself, or otherwise equal to${\displaystyle U=f_{\mathrm{c}}R=\{f_{\mathrm{c}},f_{\mathrm{d}},f_{\mathrm{v}},f_{\mathrm{h}}\}}$  (highlighted in green in the Cayley table of⁠${\displaystyle {\mathrm{D}}_4}$ ⁠). The subgroup${\displaystyle R}$  is normal, because${\displaystyle f_{\mathrm{c}}R=U=R{f}_{\mathrm{c}}}$  and similarly for the other elements of the group. (In fact, in the case of⁠${\displaystyle {\mathrm{D}}_4}$ ⁠, the cosets generated by reflections are all equal:⁠${\displaystyle f_{\mathrm{h}}R=f_{\mathrm{v}}R=f_{\mathrm{d}}R=f_{\mathrm{c}}R}$ ⁠.)

Suppose that${\displaystyle N}$  is a normal subgroup of a group⁠${\displaystyle G}$ ⁠, and$${\displaystyle G/N=\{gN\mid g\in G\}}$$ denotes its set of cosets.Then there is a unique group law on${\displaystyle G/N}$  for which the map${\displaystyle G\to G/N}$  sending each element${\displaystyle g}$  to${\displaystyle gN}$  is a homomorphism.Explicitly, the product of two cosets${\displaystyle gN}$  and${\displaystyle hN}$  is⁠${\displaystyle (gh)N}$ ⁠, the coset${\displaystyle eN=N}$  serves as the identity of⁠${\displaystyle G/N}$ ⁠, and the inverse of${\displaystyle gN}$  in the quotient group is⁠${\displaystyle (gN{)}^{-1}=\left(g^{-1}\right)N}$ ⁠.The group⁠${\displaystyle G/N}$ ⁠, read as "⁠${\displaystyle G}$ ⁠ modulo⁠${\displaystyle N}$ ⁠",^[36] is called aquotient group orfactor group.The quotient group can alternatively be characterized by auniversal property.

The elements of the quotient group${\displaystyle {\mathrm{D}}_4/R}$  are${\displaystyle R}$  and⁠${\displaystyle U=f_{\mathrm{v}}R}$ ⁠. The group operation on the quotient is shown in the table. For example,⁠${\displaystyle U\cdot U=f_{\mathrm{v}}R\cdot f_{\mathrm{v}}R=(f_{\mathrm{v}}\cdot f_{\mathrm{v}})R=R}$ ⁠. Both the subgroup${\displaystyle R=\{\mathrm{i}\mathrm{d},r_1,r_2,r_3\}}$  and the quotient${\displaystyle {\mathrm{D}}_4/R}$  are abelian, but${\displaystyle {\mathrm{D}}_4}$  is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by thesemidirect product construction;${\displaystyle {\mathrm{D}}_4}$  is an example.

Thefirst isomorphism theorem implies that anysurjective homomorphism${\displaystyle \varphi :G\to H}$  factors canonically as a quotient homomorphism followed by an isomorphism:⁠${\displaystyle G\to G/\ker \varphi \stackrel{\sim }{\to }H}$ ⁠.Surjective homomorphisms are theepimorphisms in the category of groups.

Every group is isomorphic to a quotient of afree group, in many ways.

For example, the dihedral group${\displaystyle {\mathrm{D}}_4}$  is generated by the right rotation${\displaystyle r_1}$  and the reflection${\displaystyle f_{\mathrm{v}}}$  in a vertical line (every element of${\displaystyle {\mathrm{D}}_4}$  is a finite product of copies of these and their inverses).Hence there is a surjective homomorphism⁠${\displaystyle \varphi }$ ⁠ from the free group${\displaystyle ⟨r,f⟩}$  on two generators to${\displaystyle {\mathrm{D}}_4}$  sending${\displaystyle r}$  to${\displaystyle r_1}$  and${\displaystyle f}$  to⁠${\displaystyle f_1}$ ⁠.Elements in${\displaystyle \ker \varphi }$  are calledrelations; examples include⁠${\displaystyle r^4,f^2,(r\cdot f{)}^2}$ ⁠.In fact, it turns out that${\displaystyle \ker \varphi }$  is the smallest normal subgroup of${\displaystyle ⟨r,f⟩}$  containing these three elements; in other words, all relations are consequences of these three.The quotient of the free group by this normal subgroup is denoted⁠${\displaystyle ⟨r,f\mid r^4=f^2=(r\cdot f{)}^2=1⟩}$ ⁠.This is called apresentation of${\displaystyle {\mathrm{D}}_4}$  by generators and relations, because the first isomorphism theorem for⁠${\displaystyle \varphi }$ ⁠ yields an isomorphism⁠${\displaystyle ⟨r,f\mid r^4=f^2=(r\cdot f{)}^2=1⟩\to {\mathrm{D}}_4}$ ⁠.^[37]

A presentation of a group can be used to construct theCayley graph, a graphical depiction of adiscrete group.^[38]

Examples and applications of groups abound. A starting point is the group${\displaystyle \mathbb{Z}}$  of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtainsmultiplicative groups. These groups are predecessors of important constructions inabstract algebra.

Groups are also applied in many other mathematical areas. Mathematical objects are often examined byassociating groups to them and studying the properties of the corresponding groups. For example,Henri Poincaré founded what is now calledalgebraic topology by introducing thefundamental group.^[39] By means of this connection,topological properties such asproximity andcontinuity translate into properties of groups.^[g]

Elements of the fundamental group of atopological space areequivalence classes of loops, where loops are considered equivalent if one can besmoothly deformed into another, and the group operation is "concatenation" (tracing one loop then the other). For example, as shown in the figure, if the topological space is the plane with one point removed, then loops which do not wrap around the missing point (blue)can be smoothly contracted to a single point and are the identity element of the fundamental group. A loop which wraps around the missing point${\displaystyle k}$  times cannot be deformed into a loop which wraps${\displaystyle m}$  times (with⁠${\displaystyle m\ne k}$ ⁠), because the loop cannot be smoothly deformed across the hole, so each class of loops is characterized by itswinding number around the missing point. The resulting group is isomorphic to the integers under addition.

In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background.^[h] In a similar vein,geometric group theory employs geometric concepts, for example in the study ofhyperbolic groups.^[40] Further branches crucially applying groups includealgebraic geometry and number theory.^[41]

In addition to the above theoretical applications, many practical applications of groups exist.Cryptography relies on the combination of the abstract group theory approach together withalgorithmical knowledge obtained incomputational group theory, in particular when implemented for finite groups.^[42] Applications of group theory are not restricted to mathematics; sciences such asphysics,chemistry andcomputer science benefit from the concept.

Many number systems, such as the integers and therationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known asrings and fields. Further abstract algebraic concepts such asmodules,vector spaces andalgebras also form groups.

The group of integers${\displaystyle \mathbb{Z}}$  under addition, denoted⁠${\displaystyle \left(\mathbb{Z},+\right)}$ ⁠, has been described above. The integers, with the operation of multiplication instead of addition,${\displaystyle \left(\mathbb{Z},\cdot \right)}$  donot form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example,${\displaystyle a=2}$  is an integer, but the only solution to the equation${\displaystyle a\cdot b=1}$  in this case is⁠${\displaystyle b=\frac{1}{2}}$ ⁠, which is a rational number, but not an integer. Hence not every element of${\displaystyle \mathbb{Z}}$  has a (multiplicative) inverse.^[i]

The desire for the existence of multiplicative inverses suggests consideringfractions$${\displaystyle \frac{a}{b}.}$$ 

Fractions of integers (with${\displaystyle b}$  nonzero) are known asrational numbers.^[j] The set of all such irreducible fractions is commonly denoted⁠${\displaystyle \mathbb{Q}}$ ⁠. There is still a minor obstacle for⁠${\displaystyle \left(\mathbb{Q},\cdot \right)}$ ⁠, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no${\displaystyle x}$  such that⁠${\displaystyle x\cdot 0=1}$ ⁠),${\displaystyle \left(\mathbb{Q},\cdot \right)}$  is still not a group.

However, the set of allnonzero rational numbers${\displaystyle \mathbb{Q}\setminus \left\{0\right\}=\left\{q\in \mathbb{Q}\mid q\ne 0\right\}}$  does form an abelian group under multiplication, also denoted⁠${\displaystyle {\mathbb{Q}}^{\times }}$ ⁠.^[k] Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of${\displaystyle a/b}$  is⁠${\displaystyle b/a}$ ⁠, therefore the axiom of the inverse element is satisfied.

The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – ifdivision by other than zero is possible, such as in${\displaystyle \mathbb{Q}}$  – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.^[l]

Modular arithmetic for amodulus${\displaystyle n}$  defines any two elements${\displaystyle a}$  and${\displaystyle b}$  that differ by a multiple of${\displaystyle n}$  to be equivalent, denoted by⁠${\displaystyle a\equiv b(\mathrm{mod}n)}$ ⁠. Every integer is equivalent to one of the integers from${\displaystyle 0}$  to⁠${\displaystyle n-1}$ ⁠, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalentrepresentative. Modular addition, defined in this way for the integers from${\displaystyle 0}$  to⁠${\displaystyle n-1}$ ⁠, forms a group, denoted as${\displaystyle {\mathrm{Z}}_n}$  or⁠${\displaystyle (\mathbb{Z}/n\mathbb{Z},+)}$ ⁠, with${\displaystyle 0}$  as the identity element and${\displaystyle n-a}$  as the inverse element of⁠${\displaystyle a}$ ⁠.

A familiar example is addition of hours on the face of aclock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on${\displaystyle 9}$  and is advanced${\displaystyle 4}$  hours, it ends up on⁠${\displaystyle 1}$ ⁠, as shown in the illustration. This is expressed by saying that${\displaystyle 9+4}$  is congruent to${\displaystyle 1}$  "modulo⁠${\displaystyle 12}$ ⁠" or, in symbols,$${\displaystyle 9+4\equiv 1(\mathrm{mod}12).}$$ 

For any prime number⁠${\displaystyle p}$ ⁠, there is also themultiplicative group of integers modulo⁠${\displaystyle p}$ ⁠.^[43] Its elements can be represented by${\displaystyle 1}$  to⁠${\displaystyle p-1}$ ⁠. The group operation, multiplication modulo⁠${\displaystyle p}$ ⁠, replaces the usual product by its representative, theremainder of division by⁠${\displaystyle p}$ ⁠. For example, for⁠${\displaystyle p=5}$ ⁠, the four group elements can be represented by⁠${\displaystyle 1,2,3,4}$ ⁠. In this group,⁠${\displaystyle 4\cdot 4\equiv 1mod5}$ ⁠, because the usual product${\displaystyle 16}$  is equivalent to⁠${\displaystyle 1}$ ⁠: when divided by${\displaystyle 5}$  it yields a remainder of⁠${\displaystyle 1}$ ⁠. The primality of${\displaystyle p}$  ensures that the usual product of two representatives is not divisible by⁠${\displaystyle p}$ ⁠, and therefore that the modular product is nonzero.^[m] The identity element is represented by ⁠${\displaystyle 1}$ ⁠, and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer${\displaystyle a}$  not divisible by⁠${\displaystyle p}$ ⁠, there exists an integer${\displaystyle b}$  such that$${\displaystyle a\cdot b\equiv 1(\mathrm{mod}p),}$$ that is, such that${\displaystyle p}$  evenly divides⁠${\displaystyle a\cdot b-1}$ ⁠. The inverse${\displaystyle b}$  can be found by usingBézout's identity and the fact that thegreatest common divisor${\displaystyle gcd(a,p)}$  equals ⁠${\displaystyle 1}$ ⁠.^[44] In the case${\displaystyle p=5}$  above, the inverse of the element represented by${\displaystyle 4}$  is that represented by⁠${\displaystyle 4}$ ⁠, and the inverse of the element represented by${\displaystyle 3}$  is represented by ⁠${\displaystyle 2}$ ⁠, as⁠${\displaystyle 3\cdot 2=6\equiv 1mod5}$ ⁠. Hence all group axioms are fulfilled. This example is similar to${\displaystyle \left(\mathbb{Q}\setminus \left\{0\right\},\cdot \right)}$  above: it consists of exactly those elements in the ring${\displaystyle \mathbb{Z}/p\mathbb{Z}}$  that have a multiplicative inverse.^[45] These groups, denoted⁠${\displaystyle {\mathbb{F}}_p^{\times }}$ ⁠, are crucial topublic-key cryptography.^[n]

Acyclic group is a group all of whose elements arepowers of a particular element${\displaystyle a}$ .^[46] In multiplicative notation, the elements of the group are$${\displaystyle \dots ,a^{-3},a^{-2},a^{-1},a^0,a,a^2,a^3,\dots ,}$$ where${\displaystyle a^2}$  means⁠${\displaystyle a\cdot a}$ ⁠,${\displaystyle a^{-3}}$  stands for⁠${\displaystyle a^{-1}\cdot a^{-1}\cdot a^{-1}=(a\cdot a\cdot a{)}^{-1}}$ ⁠, etc.^[o] Such an element${\displaystyle a}$  is called a generator or aprimitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as$${\displaystyle \dots ,(-a)+(-a),-a,0,a,a+a,\dots .}$$