Group (mathematics)

One of the more familiar groups is the set of integers

${\displaystyle \mathbb {Z} =\{\ldots ,-4,-3,-2,-1,0,1,2,3,4,\ldots \}}$

together with addition.[3] 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 ${\displaystyle \mathbb {Z} }$. The following properties of integer addition serve as a model for the group axioms in the definition below.

• For all integers a, 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 ⋅, that combines any two elements a and b to form an element of G, denoted a ⋅ b, in a way such that the following three requirements, known as group axioms, are satisfied:[5][6][7]

Associativity

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

Identity element

There exists an element e in G such that, for every a in G, one has e ⋅ a = a and a ⋅ e = a. Such an element is unique (see below). It is called the identity element of the group.

Inverse element

For each a in G, there exists an element b in G such that a ⋅ b = e and b ⋅ a = e, where e is the identity element. For each a, the element b is unique (see below); it is called the inverse of a and is commonly denoted a⁻¹.

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

The additive group of the field ${\displaystyle \mathbb {R} }$ is the group whose underlying set is ${\displaystyle \mathbb {R} }$ and whose operation is addition. The multiplicative 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 an additive group whenever the group operation is notated as addition; in this case, the identity is typically denoted 0,[8] 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 ⋅ b.

The definition of a group does not require that a ⋅ b = b ⋅ 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 ${\displaystyle f\circ g}$; 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.

An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set equipped with three operations, a binary operation, which is the group operation, a unary operation, which provides the inverse of its single operand, and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same.

This variant of the definition avoids existential quantifiers. It is generally preferred for computing with groups and for computer-aided proofs. This formulation exhibits groups as a variety of universal algebra. It is also useful for talking of properties of the inverse operation, as needed for defining topological groups and group objects.

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 elements of the symmetry group of the square (D₄). Vertices are identified by color or number.

id (keeping it as it is)	r1 (rotation by 90° clockwise)	r2 (rotation by 180°)	r3 (rotation by 270° clockwise)
fv (vertical reflection)	fh (horizontal reflection)	fd (diagonal reflection)	fc (counter-diagonal reflection)

• the identity operation leaving everything unchanged, denoted id;
• rotations of the square around its center by 90°, 180°, and 270° clockwise, denoted by r₁, r₂ and r₃, respectively;
• reflections about the horizontal and vertical middle line (f_v and f_h), or through the two diagonals (f_d and f_c).

These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, r₁ sends a point to its rotation 90° clockwise around the square's center, and f_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 4, denoted D₄. The underlying set of the group is the above set of symmetries, and the group operation is function composition.[9] 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 ${\displaystyle b\circ a}$ ("apply the symmetry b after performing the symmetry a"). (This is the usual notation for composition of functions.)

The group table on the right lists the results of all such compositions possible. For example, rotating by 270° clockwise (r₃) and then reflecting horizontally (f_h) is the same as performing a reflection along the diagonal (f_d). Using the above symbols, highlighted in blue in the group 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.

Composition is a binary operation. That is, ${\displaystyle a\circ b}$ is a symmetry for any two symmetries a and 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 (f_c). Indeed every other combination of two symmetries still gives a symmetry, as can be checked using the group table.

The associativity axiom deals with composing more than two symmetries: Starting with three elements a, b and c of D₄, 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 and b into a single symmetry, then to compose that symmetry with c. The other way is to first compose b and c, then to compose the resulting symmetry with 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 group table at the right:

${\displaystyle {\begin{aligned}(f_{\mathrm {d} }\circ f_{\mathrm {v} })\circ r_{2}&=r_{3}\circ r_{2}=r_{1}\\f_{\mathrm {d} }\circ (f_{\mathrm {v} }\circ r_{2})&=f_{\mathrm {d} }\circ f_{\mathrm {h} }=r_{1}.\end{aligned}}}$

The identity element is id, as, it does not changes any symmetry a when composed with it either on the left or on the right.

All symmetries have an inverse: id, the reflections f_h, f_v, f_d, f_c and the 180° rotation r₂ are their own inverse, because performing them twice brings the square back to its original orientation. The rotations r₃ and r₁ 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 irrelevant, it does matter in D₄, 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, D₄ is not abelian.

The group axioms imply that the identity element is unique: If e and f are identity elements of a group, then e = e ⋅ f = f. Therefore it is customary to speak of the identity.[29]

The group axioms also imply that the inverse of each element is unique: If a group element a has both b and c as inverses, then

b	=	b ⋅ e		since e is the identity element
	=	b ⋅ (a ⋅ c)		since c is an inverse of a, so e = a ⋅ c
	=	(b ⋅ a) ⋅ c		by associativity, which allows rearranging the parentheses
	=	e ⋅ c		since b is an inverse of a, so b ⋅ a = e
	=	c		since e is the identity element.

Therefore it is customary to speak of the inverse of an element.[29]

Given elements a and b of a group G, there is a unique solution x in G to the equation a ⋅ x = b, namely a⁻¹ ⋅ b. (One usually avoids using notations such as ${\displaystyle {\tfrac {b}{a}}}$ or b/a, unless G is abelian, because of the ambiguity of whether they mean a⁻¹ ⋅ b or b ⋅ a⁻¹.)[30] It follows that for each a in G, the function G → G given by x ↦ a ⋅ x is a bijection; it is called left multiplication by a or left translation by a.

Similarly, given a and b, the unique solution to x ⋅ a = b is b ⋅ a⁻¹. For each a, the function G → G given by x ↦ x ⋅ a is a bijection called right multiplication by a or right translation by a.

Group homomorphisms^[g] are functions that respect group structure; they may be used to relate two groups. A homomorphism from a group (G, ⋅) to a group (H, ∗) is a function φ: G → H such that

φ(a ⋅ b) = φ(a) ∗ φ(b) for all elements a, b in G.

It would be natural to require also that φ(1_G) = 1_H and φ(a^–1) = φ(a)^–1 for all a in G, but this is not necessary, because these conditions are already implied by the definition above.[31]

The identity homomorphism of a group G is the homomorphism id_G: G → G defined by g ↦ g. An inverse homomorphism of a homomorphism φ: G → H is a homomorphism Φ: H → G such that Φ ∘ φ = id_G and φ ∘ Φ = id_H, i.e., such that Φ(φ(g)) = g for all g in G and φ(Φ(h)) = 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 φ: G → H; in this case, H can be obtained from G simply by renaming its elements according to the function φ; then any statement true for G is true for H, provided that any specific elements mentioned in the statement are also renamed.

Informally, a subgroup is a group H contained within a bigger one, G.[32] Concretely, the identity element of G is contained in H, and whenever h₁ and h₂ are in H, then so are h₁ ⋅ h₂ and h₁⁻¹, so the elements of H, equipped with the group operation on G restricted to H, indeed form a group.

In the example above, the identity and the rotations constitute a subgroup R = {id, r₁, r₂, r₃}, highlighted in red in the group table above: 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° (note that rotation in the opposite direction is not defined). The subgroup test is 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⁻¹h ∈ H for all elements g, h ∈ H. Knowing the subgroups is important in understanding the group as a whole.^[d]

Given any subset S of a group G, the subgroup generated by S consists of products of elements of S and their inverses. It is the smallest subgroup of G containing S.[33] In the introductory example above, the subgroup generated by r₂ and f_v consists of these two elements, the identity element id and f_h = f_v ⋅ r₂. 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 D₄ above, once a reflection is performed, the square never gets back to the r₂ configuration by just applying the rotation operations (and no further reflections), i.e., the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup H defines left and right cosets, which can be thought of as translations of H by arbitrary group elements g. In symbolic terms, the left and right cosets of H containing g are

gH = {g ⋅ h : h ∈ H} and Hg = {h ⋅ g : h ∈ H}, respectively.[34]

The left cosets of any subgroup H form a partition of G; that is, the union of all left cosets is equal to G and two left cosets are either equal or have an empty intersection.[35] The first case g₁H = g₂H happens precisely when g₁⁻¹ ⋅ g₂ ∈ H, i.e., if the two elements differ by an element of H. Similar considerations apply to the right cosets of H. The left and right cosets of H may or may not be equal. If they are, i.e., for all g in G, gH = Hg, then H is said to be a normal subgroup.

In D₄, the introductory symmetry group, the left cosets gR of the subgroup R consisting of the rotations are either equal to R, if g is an element of R itself, or otherwise equal to U = f_cR = {f_c, f_v, f_d, f_h} (highlighted in green). The subgroup R is also normal, because f_cR = U = Rf_c and similarly for any element other than f_c. (In fact, in the case of D₄, observe that all such cosets are equal, such that f_hR = f_vR = f_dR = f_cR.)

In some situations the set of cosets of a subgroup can be endowed with a group law, giving a quotient group or factor group. For this to be possible, the subgroup has to be normal. Given any normal subgroup N, the quotient group is defined by

G / N = {gN, g ∈ G}, "G modulo N".[36]

This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) ⋅ (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to any element g its coset gN be a group homomorphism, or by general abstract considerations called universal properties. The coset eN = N serves as the identity in this group, and the inverse of gN in the quotient group is (gN)⁻¹ = (g⁻¹)N.^[e]

The elements of the quotient group D₄ / R are R itself, which represents the identity, and U = f_vR. The group operation on the quotient is shown at the right. For example, U ⋅ U = f_vR ⋅ f_vR = (f_v ⋅ f_v)R = R. Both the subgroup R = {id, r₁, r₂, r₃}, as well as the corresponding quotient are abelian, whereas D₄ is not abelian. Building bigger groups by smaller ones, such as D₄ from its subgroup R and the quotient D₄ / R is abstracted by a notion called semidirect product.

Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D₄, for example, can be generated by two elements r and f (for example, r = r₁, the right rotation and f = f_v the vertical (or any other) reflection), which means that every symmetry of the square is a finite composition of these two symmetries or their inverses. Together with the relations

r ⁴ = f ² = (r ⋅ f)² = 1,[37]

the group is completely described. A presentation of a group can also be used to construct the Cayley graph, a device used to graphically capture discrete groups.

Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e., any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / N.^[y] Interpreting subgroup and quotients in light of these homomorphisms emphasizes the structural concept inherent to these definitions alluded to in the introduction. In general, homomorphisms are neither injective nor surjective. Kernel and image of group homomorphisms and the first isomorphism theorem address this phenomenon.

Many number systems, such as the integers and the rationals 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 as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras 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, ${\displaystyle \left(\mathbb {Z} ,\cdot \right)}$ do not form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, a = 2 is an integer, but the only solution to the equation a · b = 1 in this case is b = 1/2, which is a rational number, but not an integer. Hence not every element of ${\displaystyle \mathbb {Z} }$ has a (multiplicative) inverse.^[k]

The desire for the existence of multiplicative inverses suggests considering fractions

${\displaystyle {\frac {a}{b}}.}$

Fractions of integers (with b nonzero) are known as rational numbers.^[l] 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 the rational number 0 does not have a multiplicative inverse (i.e., there is no x such that x · 0 = 1), ${\displaystyle \left(\mathbb {Q} ,\cdot \right)}$ is still not a group.

However, the set of all nonzero rational numbers ${\displaystyle \mathbb {Q} \setminus \left\{0\right\}=\left\{q\in \mathbb {Q} \mid q\neq 0\right\}}$ does form an abelian group under multiplication, generally denoted ${\displaystyle \mathbb {Q} ^{*}}$.^[m] 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 a/b is b/a, therefore the axiom of the inverse element is satisfied.

The rational numbers (including 0) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and—if division 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.^[n]

The hours on a clock form a group that uses addition modulo 12. Here 9 + 4 = 1.

In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0 is the identity element. This is familiar from the addition of hours on the face of a clock: if the hour hand is on 9 and is advanced 4 hours, it ends up on 1, as shown at the right. This is expressed by saying that 9 + 4 equals 1 "modulo 12" or, in symbols,

9 + 4 ≡ 1 modulo 12.

The group of integers modulo n is written ${\displaystyle \mathbb {Z} _{n}}$ or ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$.

For any prime number p, there is also the multiplicative group of integers modulo p.[42] Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, there are four group elements 1, 2, 3, 4. In this group, 4 · 4 = 1, because the usual product 16 is equivalent to 1, which divided by 5 yields a remainder of 1. for 5 divides 16 − 1 = 15, denoted

16 ≡ 1 (mod 5).

The primality of p ensures that the product of two integers neither of which is divisible by p is not divisible by p either, hence the indicated set of classes is closed under multiplication.^[o] The identity element is 1, as usual for a multiplicative group, and the associativity follows from the corresponding property of integers. Finally, the inverse element axiom requires that given an integer a not divisible by p, there exists an integer b such that

a · b ≡ 1 (mod p), i.e., p divides the difference a · b − 1.

The inverse b can be found by using Bézout's identity and the fact that the greatest common divisor gcd(a, p) equals 1.[43] In the case p = 5 above, the inverse of 4 is 4, and the inverse of 3 is 2, as 3 · 2 = 6 ≡ 1 (mod 5). Hence all group axioms are fulfilled. Actually, this example is similar to ${\displaystyle \left(\mathbb {Q} \setminus \left\{0\right\},\cdot \right)}$ above: it consists of exactly those elements in ${\displaystyle \mathbb {Z} /p\mathbb {Z} }$ that have a multiplicative inverse.[44] These groups are denoted F_p^×. They are crucial to public-key cryptography.^[p]

The 6th complex roots of unity form a cyclic group. z is a primitive element, but z² is not, because the odd powers of z are not a power of z².

A cyclic group is a group all of whose elements are powers of a particular element a.[45] In multiplicative notation, the elements of the group are:

..., a⁻³, a⁻², a⁻¹, a⁰ = e, a, a², a³, ...,

where a² means a ⋅ a, and a⁻³ stands for a⁻¹ ⋅ a⁻¹ ⋅ a⁻¹ = (a ⋅ a ⋅ a)⁻¹ etc.^[h] Such an element a is called a generator or a primitive 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

..., −a−a, −a, 0, a, a+a, ...

In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by complex numbers z satisfying zⁿ = 1. These numbers can be visualized as the vertices on a regular n-gon, as shown in blue at the right for n = 6. The group operation is multiplication of complex numbers. In the picture, multiplying with z corresponds to a counter-clockwise rotation by 60°.[46] Using some field theory, the group F_p^× can be shown to be cyclic: for example, if p = 5, 3 is a generator since 3¹ = 3, 3² = 9 ≡ 4, 3³ ≡ 2, and 3⁴ ≡ 1.

Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element a, all the powers of a are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to (Z, +), the group of integers under addition introduced above.[47] As these two prototypes are both abelian, so is any cyclic group.

The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.[48]

Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions.[49] Conceptually, group theory can be thought of as the study of symmetry.^[t] Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object X if every group element can be associated to some operation on X and the composition of these operations follows the group law. In the rightmost example below, an element of order 7 of the (2,3,7) triangle group acts on the tiling by permuting the highlighted warped triangles (and the other ones, too). By a group action, the group pattern is connected to the structure of the object being acted on.

Rotations and reflections form the symmetry group of a great icosahedron.

In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties.[50] For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.

Not only are groups useful to assess the implications of symmetries in molecules, but surprisingly they also predict that molecules sometimes can change symmetry. The Jahn-Teller effect is a distortion of a molecule of high symmetry when it adopts a particular ground state of lower symmetry from a set of possible ground states that are related to each other by the symmetry operations of the molecule.[51][52]

Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.[53]

Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.

Buckminsterfullerene displays icosahedral symmetry, though the double bonds reduce this to pyritohedral symmetry.	Ammonia, NH3. Its symmetry group is of order 6, generated by a 120° rotation and a reflection.	Cubane C8H8 features octahedral symmetry.	Hexaaquacopper(II) complex ion, [Cu(OH2)6]2+. Compared to a perfectly symmetrical shape, the molecule is vertically dilated by about 22% (Jahn-Teller effect).	The (2,3,7) triangle group, a hyperbolic group, acts on this tiling of the hyperbolic plane.

Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players.[54] Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved.^[u] Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.[55]

Two vectors (the left illustration) multiplied by matrices (the middle and right illustrations). The middle illustration represents a clockwise rotation by 90°, while the right-most one stretches the x-coordinate by factor 2.

Matrix groups consist of matrices together with matrix multiplication. The general linear group GL(n, R) consists of all invertible n-by-n matrices with real entries.[56] Its subgroups are referred to as matrix groups or linear groups. The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group SO(n). It describes all possible rotations in n dimensions. Via Euler angles, rotation matrices are used in computer graphics.[57]

Representation theory is both an application of the group concept and important for a deeper understanding of groups.[58][59] It studies the group by its group actions on other spaces. A broad class of group representations are linear representations, i.e., the group is acting on a vector space, such as the three-dimensional Euclidean space R³. A representation of G on an n-dimensional real vector space is simply a group homomorphism

ρ: G → GL(n, R)

from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.^[w]

Given a group action, this gives further means to study the object being acted on.^[x] On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.[58][60]

Galois groups were developed to help solve polynomial equations by capturing their symmetry features.[61][62] For example, the solutions of the quadratic equation ax² + bx + c = 0 are given by

${\displaystyle x={\frac {-b\pm {\sqrt {b^{2}-4ac}}}{2a}}.}$

Exchanging "+" and "−" in the expression, i.e., permuting the two solutions of the equation can be viewed as a (very simple) group operation. Similar formulae are known for cubic and quartic equations, but do not exist in general for degree 5 and higher.[63] Abstract properties of Galois groups associated with polynomials (in particular their solvability) give a criterion for polynomials that have all their solutions expressible by radicals, i.e., solutions expressible using solely addition, multiplication, and roots similar to the formula above.[64]

The problem can be dealt with by shifting to field theory and considering the splitting field of a polynomial. Modern Galois theory generalizes the above type of Galois groups to field extensions and establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.

Mathematicians often strive for a complete classification (or list) of a mathematical notion. In the context of finite groups, this aim leads to difficult mathematics. According to Lagrange's theorem, finite groups of order p, a prime number, are necessarily cyclic (abelian) groups Z_p. Groups of order p² can also be shown to be abelian, a statement which does not generalize to order p³, as the non-abelian group D₄ of order 8 = 2³ above shows.[66] Computer algebra systems can be used to list small groups, but there is no classification of all finite groups.^[q] An intermediate step is the classification of finite simple groups.^[r] A nontrivial group is called simple if its only normal subgroups are the trivial group and the group itself.^[s] The Jordan–Hölder theorem exhibits finite simple groups as the building blocks for all finite groups.[67] Listing all finite simple groups was a major achievement in contemporary group theory. 1998 Fields Medal winner Richard Borcherds succeeded in proving the monstrous moonshine conjectures, a surprising and deep relation between the largest finite simple sporadic group (the "monster group") and certain modular functions, a piece of classical complex analysis, and string theory, a theory supposed to unify the description of many physical phenomena.[68]

The unit circle in the complex plane under complex multiplication is a Lie group and, therefore, a topological group. It is topological since complex multiplication and division are continuous. It is a manifold and thus a Lie group, because every small piece, such as the red arc in the figure, looks like a part of the real line (shown at the bottom).

Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, g ⋅ h and g⁻¹ must not vary wildly if g and h vary only a little. Such groups are called topological groups, and they are the group objects in the category of topological spaces.[70] The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of p-adic numbers. These examples are locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory.[71] Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory.[72] A generalization used in algebraic geometry is the étale fundamental group.[73]

A Lie group is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension.[74] Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth.

A standard example is the general linear group introduced above: it is an open subset of the space of all n-by-n matrices, because it is given by the inequality

det (A) ≠ 0,

where A denotes an n-by-n matrix.[75]

Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities.[76] Rotation, as well as translations in space and time are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description.^[v] Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of spacetime in special relativity.[77] The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories.[78] Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory.[79]

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