Group (mathematics)

In mathematics, a group is a set with an operation that combines any two elements of the set to produce a third element within the same set and the following conditions must hold: the operation is associative, it has an identity element, and every element of the set has an inverse element. For example, the integers with the addition operation form a group.

The concept of a group was elaborated for handling, in a unified way, many mathematical structures such as numbers, geometric shapes and polynomial 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.

In geometry, groups arise naturally in the study of symmetries and geometric transformations: the symmetries of an object form a group, called the symmetry 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 the Standard Model of particle physics. The Poincaré group is a Lie group consisting of the symmetries of spacetime in special relativity. Point groups describe symmetry in molecular chemistry.

The concept of a group arose in the study of polynomial equations. Évariste Galois, in the 1830s, introduced the term group (French: groupe) for the symmetry group of the roots of an equation, now called a Galois group. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group 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 as subgroups, quotient groups and simple 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 of representation theory (that is, through the representations of the group) and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory.

One of the more familiar groups is the set of integers \[\Z = \{\ldots,-4,-3,-2,-1,0,1,2,3,4,\ldots\}\] together with addition. For any two integers \(a\) and ⁠\(b\)⁠, the sum \(a+b\) is also an integer; this closure property says that \(+\) is a binary operation on ⁠\(\Z\)⁠. The following properties of integer addition serve as a model for the group axioms in the definition below.

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

The integers, together with the operation ⁠\(+\)⁠, 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 set \(G\) together with a binary operation on ⁠\(G\)⁠, here denoted "⁠\(\cdot\)⁠", that combines any two elements \(a\) and \(b\) of \(G\) to form an element of ⁠\(G\)⁠, denoted ⁠\(a\cdot b\)⁠, such that the following three requirements, known as group axioms, are satisfied:

Associativity

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

Identity element

There exists an element \(e\) in \(G\) such that, for every \(a\) in ⁠\(G\)⁠, one has ⁠\(e\cdot a=a\)⁠ and ⁠\(a\cdot e=a\)⁠.

Such an element \(e\) is unique. It is called the identity element (or sometimes neutral element) of the group.

Inverse element

For each \(a\) in ⁠\(G\)⁠, there exists an element \(b\) in \(G\) such that \(a\cdot b=e\) and ⁠\(b\cdot a=e\)⁠, where \(e\) is the identity element.

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

Note: Uniqueness of the identity and uniqueness of inverse elements are not part of the axioms; they are consequences of the three axioms.

Formally, a group is an ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the underlying set of the group, and the operation is called the group operation or the group law.

A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse 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 of real numbers ⁠\(\R\)⁠, which has the operations of addition \(a+b\) and multiplication ⁠\(ab\)⁠. Formally, \(\R\) is a set, \((\R,+)\) is a group, and \((\R,+,\cdot)\) is a field. But it is common to write \(\R\) to denote any of these three objects.

The additive group of the field \(\R\) is the group whose underlying set is \(\R\) and whose operation is addition. The multiplicative group of the field \(\R\) is the group \(\R^{\times}\) whose underlying set is the set of nonzero real numbers \(\R \smallsetminus \{0\}\) and whose operation is multiplication.

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

The definition of a group does not require that \(a\cdot b=b\cdot a\) for all elements \(a\) and \(b\) in ⁠\(G\)⁠. If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian 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 are functions, the operation is often function composition ⁠\(f\circ g\)⁠; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups, and automorphism groups, the symbol \(\circ\) is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. A square has eight symmetries. These are:

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

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, \(r_1\) sends a point to its rotation 90° clockwise around the square's center, and \(f_{\mathrm{h}}\) 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 the dihedral group of degree four, denoted ⁠\(\mathrm{D}_4\)⁠. The underlying set of the group is the above set of symmetries, and the group operation is function composition. 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 \(a\) and then \(b\) is written symbolically from right to left as \(b\circ a\) ("apply the symmetry \(b\) after performing the symmetry ⁠\(a\)⁠"). This is the usual notation for the composition of functions.

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

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

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

Condensed: the full section is in Wikipedia.

The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation \(\theta^n=1\) (1854) gives the first abstract definition of a finite group.

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.

The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss's number-theoretical work Disquisitiones Arithmeticae (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers.

The convergence of these various sources into a uniform theory of groups started with Jordan (1870)'s Traité des substitutions et des équations algébriques. 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. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups and wrote the first book about group theory in the English language: Theory of Groups of Finite Order, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.

Condensed: the full section is in Wikipedia.

Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under elementary group theory. For example, repeated applications of the associativity axiom show that the unambiguity of \[a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot(b\cdot c)\] generalizes to more than three factors (for example, \(a\cdot b\cdot c\cdot d\) is also unambiguous). Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.

The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements \(e\) and \(f\) of a group are equal, because the group axioms imply ⁠\(e=e\cdot f=f\)⁠. It is thus customary to speak of the identity element of the group.

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

\(\begin{align} b &= b\cdot e && \text{(}e \text { is the identity element)}\\ &= b\cdot (a \cdot c) && \text{(}c \text { and } a \text{ are inverses of each other)}\\ &= (b\cdot a) \cdot c && \text{(associativity)}\\ &= e \cdot c && \text{(}b \text { is an inverse of } a\text{)}\\ &= c && \text{(}e \text { is the identity element and } b=c\text{)} \end{align}\)

Therefore, it is customary to speak of the inverse of an element.

Given elements \(a\) and \(b\) of a group \(G\), there is a unique solution \(x\) in \(G\) to the equation \(a\cdot x=b\), namely \(a^{-1}\cdot b\). It follows that for each \(a\) in \(G\), the function \(G\to G\) that maps each \(x\) to \(a\cdot x\) is a bijection; it is called left multiplication by \(a\) or left translation by \(a\).

Similarly, given \(a\) and \(b\) in \(G\), the unique solution \(x\) to \(x\cdot a=b\) is \(b\cdot a^{-1}\). For each \(a\), the function \(G\to G\) that maps each \(x\) to \(x\cdot a\) is a bijection called right multiplication by \(a\) or right translation by \(a\).

The group axioms for identity and inverses may be "weakened" to assert only the existence of a left identity and left inverses. From these one-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.

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

\(\begin{align} f \cdot f^{-1} &=e \cdot (f \cdot f^{-1}) && \text{(left identity)}\\ &=((f^{-1})^{-1} \cdot f^{-1}) \cdot (f \cdot f^{-1}) && \text{(left inverse)}\\ &=(f^{-1})^{-1} \cdot ((f^{-1} \cdot f) \cdot f^{-1}) && \text{(associativity)}\\ &=(f^{-1})^{-1} \cdot (e \cdot f^{-1}) && \text{(left inverse)}\\ &=(f^{-1})^{-1} \cdot f^{-1} && \text{(left identity)}\\ &=e && \text{(left inverse)} \end{align}\)

Similarly, the left identity is also a right identity:

\(\begin{align} f\cdot e &= f \cdot ( f^{-1} \cdot f) && \text{(left inverse)}\\ &= (f \cdot f^{-1}) \cdot f && \text{(associativity)}\\ &= e \cdot f && \text{(right inverse)}\\ &= f && \text{(left identity)} \end{align}\)

These results do not hold if any of these axioms (associativity, existence of left identity and existence of left inverse) is removed. For a structure with a looser definition (like a semigroup) 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 a left identity and a right inverse (or vice versa) is not sufficient to define a group. For example, consider the set \(G = \{ e,f \}\) with the operator \(\,\!\cdot\) satisfying \(e \cdot e = f \cdot e = e\) and ⁠\(e \cdot f = f \cdot f = f\)⁠. This structure does have a left identity (namely, ⁠\(e\)⁠), and each element has a right inverse (which is \(e\) 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 applied). However, \(( G , \cdot )\) is not a group, since it lacks a right identity.

When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation. When studying groups, one uses instead subgroups, homomorphisms, and quotient groups. These are the analogues that take the group structure into account.

Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A homomorphism from a group \((G,\cdot)\) to a group \((H,*)\) is a function \(\varphi : G\to H\) such that

It would be natural to require also that \(\varphi\) respect identities, ⁠\(\varphi(1_G)=1_H\)⁠, and inverses, \(\varphi(a^{-1})=\varphi(a)^{-1}\) for all \(a\) in ⁠\(G\)⁠. 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.

The identity homomorphism of a group \(G\) is the homomorphism \(\iota_G : G\to G\) that maps each element of \(G\) to itself. An inverse homomorphism of a homomorphism \(\varphi : G\to H\) is a homomorphism \(\psi : H\to G\) such that \(\psi\circ\varphi=\iota_G\) and ⁠\(\varphi\circ\psi=\iota_H\)⁠, that is, such that \(\psi\bigl(\varphi(g)\bigr)=g\) for all \(g\) in \(G\) and such that \(\varphi\bigl(\psi(h)\bigr)=h\) for all \(h\) in ⁠\(H\)⁠. An isomorphism is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups \(G\) and \(H\) are called isomorphic if there exists an isomorphism ⁠\(\varphi : G\to H\)⁠. In this case, \(H\) can be obtained from \(G\) simply by renaming its elements according to the function ⁠\(\varphi\)⁠; then any statement true for \(G\) is true for ⁠\(H\)⁠, provided that any specific elements mentioned in the statement are also renamed.

The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.

An injective homomorphism \(\phi : G' \to G\) factors canonically as an isomorphism followed by an inclusion, \(G' \;\stackrel{\sim}{\to}\; H \hookrightarrow G\) for some subgroup ⁠\(H\)⁠ of ⁠\(G\)⁠. Injective homomorphisms are the monomorphisms in the category of groups.

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

In the example of symmetries of a square, the identity and the rotations constitute a subgroup ⁠\(R=\{\mathrm{id},r_1,r_2,r_3\}\)⁠, 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°. The subgroup test provides a necessary and sufficient condition for a nonempty subset ⁠\(H\)⁠ of a group ⁠\(G\)⁠ to be a subgroup: it is sufficient to check that \(g^{-1}\cdot h\in H\) for all elements \(g\) and \(h\) in ⁠\(H\)⁠. Knowing a group's subgroups is important in understanding the group as a whole.

Given any subset \(S\) of a group ⁠\(G\)⁠, the subgroup generated by \(S\) consists of all products of elements of \(S\) and their inverses. It is the smallest subgroup of \(G\) containing ⁠\(S\)⁠. In the example of symmetries of a square, the subgroup generated by \(r_2\) and \(f_{\mathrm{v}}\) consists of these two elements, the identity element ⁠\(\mathrm{id}\)⁠, and the element ⁠\(f_{\mathrm{h} }=f_{\mathrm{v} }\cdot r_2\)⁠. 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.