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.
Definition and illustration
First example: the integers One of the more familiar groups is the set of integers
Z = { … , − 4 , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , 4 , … } {\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}
together with addition. For any two integers a {\displaystyle a} and b {\displaystyle b} , the sum a + b {\displaystyle a+b} is also an integer; this closure property says that + {\displaystyle +} is a binary operation on Z {\displaystyle \mathbb {Z} } . The following properties of integer addition serve as a model for the group axioms in the definition below.
For all integers a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} , one has ( a + b ) + c = a + ( b + c ) {\displaystyle (a+b)+c=a+(b+c)} . Expressed in words, adding a {\displaystyle a} to b {\displaystyle b} first, and then adding the result to c {\displaystyle c} gives the same final result as adding a {\displaystyle a} to the sum of b {\displaystyle b} and c {\displaystyle c} . This property is known as associativity. If a {\displaystyle a} is any integer, then 0 + a = a {\displaystyle 0+a=a} and a + 0 = a {\displaystyle 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 {\displaystyle a} , there is an integer b {\displaystyle b} such that a + b = 0 {\displaystyle a+b=0} and b + a = 0 {\displaystyle b+a=0} . The integer b {\displaystyle b} is called the inverse element of the integer a {\displaystyle a} and is denoted − a {\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.
Definition A group is a set G {\displaystyle G} together with a binary operation on G {\displaystyle G} , here denoted " ⋅ {\displaystyle \cdot } ", that combines any two elements a {\displaystyle a} and b {\displaystyle b} of G {\displaystyle G} to form an element of G {\displaystyle G} , denoted a ⋅ b {\displaystyle a\cdot b} , such that the following three requirements, known as group axioms, are satisfied:
Associativity For all a {\displaystyle a} , b {\displaystyle b} , c {\displaystyle c} in G {\displaystyle G} , one has ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle (a\cdot b)\cdot c=a\cdot (b\cdot c)} . Identity element There exists an element e {\displaystyle e} in G {\displaystyle G} such that, for every a {\displaystyle a} in G {\displaystyle G} , one has e ⋅ a = a {\displaystyle e\cdot a=a} and a ⋅ e = a {\displaystyle a\cdot e=a} . Such an element e {\displaystyle e} is unique. It is called the identity element (or sometimes neutral element) of the group. Inverse element For each a {\displaystyle a} in G {\displaystyle G} , there exists an element b {\displaystyle b} in G {\displaystyle G} such that a ⋅ b = e {\displaystyle a\cdot b=e} and b ⋅ a = e {\displaystyle b\cdot a=e} , where e {\displaystyle e} is the identity element. For each a {\displaystyle a} , the element b {\displaystyle b} is unique; it is called the inverse of a {\displaystyle a} and is commonly denoted a − 1 {\displaystyle 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.
Notation and terminology 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 {\displaystyle \mathbb {R} } , which has the operations of addition a + b {\displaystyle a+b} and multiplication a b {\displaystyle ab} . Formally, R {\displaystyle \mathbb {R} } is a set, ( R , + ) {\displaystyle (\mathbb {R} ,+)} is a group, and ( R , + , ⋅ ) {\displaystyle (\mathbb {R} ,+,\cdot )} is a field. But it is common to write R {\displaystyle \mathbb {R} } to denote any of these three objects. The additive group of the field R {\displaystyle \mathbb {R} } is the group whose underlying set is R {\displaystyle \mathbb {R} } and whose operation is addition. The multiplicative group of the field R {\displaystyle \mathbb {R} } is the group R × {\displaystyle \mathbb {R} ^{\times }} whose underlying set is the set of nonzero real numbers R ∖ { 0 } {\displaystyle \mathbb {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 {\displaystyle 0} , and the inverse of an element x {\displaystyle x} is denoted − x {\displaystyle -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 {\displaystyle 1} , and the inverse of an element x {\displaystyle x} is denoted x − 1 {\displaystyle x^{-1}} . In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, a b {\displaystyle ab} instead of a ⋅ b {\displaystyle a\cdot b} . The definition of a group does not require that a ⋅ b = b ⋅ a {\displaystyle a\cdot b=b\cdot a} for all elements a {\displaystyle a} and b {\displaystyle b} in G {\displaystyle 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 ∘ g {\displaystyle 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 ∘ {\displaystyle \circ } is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.
Second example: a symmetry group
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 id {\displaystyle \operatorname {id} } ; rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by r 1 {\displaystyle r_{1}} , r 2 {\displaystyle r_{2}} and r 3 {\displaystyle r_{3}} , respectively; reflections about the horizontal and vertical middle line ( f v {\displaystyle f_{\mathrm {v} }} and f h {\displaystyle f_{\mathrm {h} }} ), or through the two diagonals ( f d {\displaystyle f_{\mathrm {d} }} and f c {\displaystyle 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 {\displaystyle r_{1}} sends a point to its rotation 90° clockwise around the square's center, and f h {\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 the dihedral group of degree four, denoted D 4 {\displaystyle \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 {\displaystyle a} and then b {\displaystyle b} is written symbolically from right to left as b ∘ a {\displaystyle b\circ a} ("apply the symmetry b {\displaystyle b} after performing the symmetry a {\displaystyle 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 {\displaystyle r_{3}} ) and then reflecting horizontally ( f h {\displaystyle f_{\mathrm {h} }} ) is the same as performing a reflection along the diagonal ( f d {\displaystyle f_{\mathrm {d} }} ). Using the above symbols, highlighted in blue in the Cayley table:
f h ∘ r 3 = f d . {\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, a ∘ b {\displaystyle a\circ b} is a symmetry for any two symmetries a {\displaystyle a} and b {\displaystyle b} . For example, r 3 ∘ f h = f c . {\displaystyle r_{3}\circ f_{\mathrm {h} }=f_{\mathrm {c} }.} That is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal ( f c {\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 a {\displaystyle a} , b {\displaystyle b} and c {\displaystyle c} of D 4 {\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 a {\displaystyle a} and b {\displaystyle b} into a single symmetry, then to compose that symmetry with c {\displaystyle c} . The other way is to first compose b {\displaystyle b} and c {\displaystyle c} , then to compose the resulting symmetry with a {\displaystyle a} . These two ways must always give the same result, that is, ( a ∘ b ) ∘ c = a ∘ ( b ∘ c ) . {\displaystyle (a\circ b)\circ c=a\circ (b\circ c).} For example, ( f d ∘ f v ) ∘ r 2 = f d ∘ ( f v ∘ r 2 ) {\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: ( f d ∘ f v ) ∘ r 2 = r 3 ∘ r 2 = r 1 f d ∘ ( f v ∘ r 2 ) = f d ∘ f h = r 1 . {\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}
Identity element: The identity element is i d {\displaystyle \mathrm {id} } , as it does not change any symmetry a {\displaystyle a} when composed with it either on the left or on the right. Inverse element: Each symmetry has an inverse: i d {\displaystyle \mathrm {id} } , the reflections f h {\displaystyle f_{\mathrm {h} }} , f v {\displaystyle f_{\mathrm {v} }} , f d {\displaystyle f_{\mathrm {d} }} , f c {\displaystyle f_{\mathrm {c} }} and the 180° rotation r 2 {\displaystyle r_{2}} are their own inverse, because performing them twice brings the square back to its original orientation. The rotations r 3 {\displaystyle r_{3}} and r 1 {\displaystyle r_{1}} are each other's inverses, because rotating 90° and then rotating 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified in the table. In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in D 4 {\displaystyle \mathrm {D} _{4}} , as, for example, f h ∘ r 1 = f c {\displaystyle f_{\mathrm {h} }\circ r_{1}=f_{\mathrm {c} }} but r 1 ∘ f h = f d {\displaystyle r_{1}\circ f_{\mathrm {h} }=f_{\mathrm {d} }} . In other words, D 4 {\displaystyle \mathrm {D} _{4}} is not abelian.
History
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 θ n = 1 {\displaystyle \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. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the examples below illustrate.
Elementary consequences of the group axioms 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 ⋅ b ⋅ c = ( a ⋅ b ) ⋅ c = a ⋅ ( b ⋅ c ) {\displaystyle a\cdot b\cdot c=(a\cdot b)\cdot c=a\cdot (b\cdot c)}
generalizes to more than three factors (for example, a ⋅ b ⋅ c ⋅ d {\displaystyle 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.
Uniqueness of identity element The group axioms imply that the identity element is unique; that is, there exists only one identity element: any two identity elements e {\displaystyle e} and f {\displaystyle f} of a group are equal, because the group axioms imply e = e ⋅ f = f {\displaystyle e=e\cdot f=f} . It is thus customary to speak of the identity element of the group.
Uniqueness of inverses The group axioms also imply that the inverse of each element is unique. Let a group element a {\displaystyle a} have both b {\displaystyle b} and c {\displaystyle c} as inverses. Then
b = b ⋅ e ( e is the identity element) = b ⋅ ( a ⋅ c ) ( c and a are inverses of each other) = ( b ⋅ a ) ⋅ c (associativity) = e ⋅ c ( b is an inverse of a ) = c ( e is the identity element and b = c ) {\displaystyle {\begin{aligned}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{aligned}}}
Therefore, it is customary to speak of the inverse of an element.
Division Given elements a {\displaystyle a} and b {\displaystyle b} of a group G {\displaystyle G} , there is a unique solution x {\displaystyle x} in G {\displaystyle G} to the equation a ⋅ x = b {\displaystyle a\cdot x=b} , namely a − 1 ⋅ b {\displaystyle a^{-1}\cdot b} . It follows that for each a {\displaystyle a} in G {\displaystyle G} , the function G → G {\displaystyle G\to G} that maps each x {\displaystyle x} to a ⋅ x {\displaystyle a\cdot x} is a bijection; it is called left multiplication by a {\displaystyle a} or left translation by a {\displaystyle a} . Similarly, given a {\displaystyle a} and b {\displaystyle b} in G {\displaystyle G} , the unique solution x {\displaystyle x} to x ⋅ a = b {\displaystyle x\cdot a=b} is b ⋅ a − 1 {\displaystyle b\cdot a^{-1}} . For each a {\displaystyle a} , the function G → G {\displaystyle G\to G} that maps each x {\displaystyle x} to x ⋅ a {\displaystyle x\cdot a} is a bijection called right multiplication by a {\displaystyle a} or right translation by a {\displaystyle a} .
Equivalent definition with relaxed axioms 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 {\displaystyle e} (that is, e ⋅ f = f {\displaystyle e\cdot f=f} ) and a left inverse f − 1 {\displaystyle f^{-1}} for each element f {\displaystyle f} (that is, f − 1 ⋅ f = e {\displaystyle 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
f ⋅ f − 1 = e ⋅ ( f ⋅ f − 1 ) (left identity)