Group (mathematics)

A group is one of the fundamental objects of study in the field of mathematics known as abstract algebra. The branch of algebra that studies groups is called group theory. Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces may be defined concisely in terms of groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics. Furthermore, their ability to represent geometric transformations finds applications in chemistry, computer graphics, and other fields.

Many investigated structures in mathematics turn out to be groups. These include familiar number systems, such as: the integers, the rational numbers, the real numbers, and the complex numbers under addition, as well as the non-zero rationals, reals, and complex numbers under multiplication. Other important examples are: the group of non-singular matrices under multiplication, and the group of invertible functions under composition. Group theory allows for the properties of such structures to be investigated in a general setting.

Definition

A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:

Closure: For all a, b in G, the result of a * b is also in G.

Associativity: For all a, b and c in G, (a * b) * c = a * (b * c).

Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a.

Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element.

Some texts omit the explicit requirement of closure, since the closure of the group follows from the definition of a binary operation.

Using the identity element property, it can be shown that a group has exactly one identity element. See the proof below.

The inverse of an element a can also be shown to be unique, and it is usually written a^-1 (but see the notation below for additively written groups).

A group (G, *) is often denoted simply G where there is no ambiguity as to what the operation is.

Illustration of definition

An example will explain some properties of groups. Consider a square. We are interested in the symmetries of the square. There are the following types of symmetries:

• rotation about 90°, 180° and 270° (clockwise). We will write these symmetries as rot_90°, rot_180°, and rot_270°, respectively. Note that the counter-clockwise rotations are included here, for example rotating 270° clockwise is equal to rotating 90° counter-clockwise.
• reflection along the vertical or horizontal middle line, or along the two diagonals. Let us write the reflections as ref_V, ref_H, ref_D1 and ref_D2, respectively.
• Finally, the identical operation id leaving everything unchanged is also a symmetry.

All of them keep the shape of the square unchanged. (In the images, the vertices are colored only for making clear the operations).

This set G of symmetries is an example for a group. This means the following:

• Two symmetries can be composed, i.e. given two symmetries a and b, we can first perform a and then b and the result will still be a symmetry. We write the result b * a (meaning "b after a"). For example, rotating by 270° and then rotating by 180° equals a rotation by 90°, i.e. using the above symbols, we have

rot_180° * rot_270° = rot_90°.

In a more formal language, G is endowed with a binary operation *, i.e. any two elements can be composed to a third element.

Applied to this example group, the definition reads:

1. Associativity: given three elements a, b and c of G, (a * b) * c = a * (b * c).
2. Identity element: There exists an element e in G such that for all a in G, e * a = a * e = a. In the example, e is just the symmetry which leaves everything unchanged.
3. Inverse element: For each a in G, there exists an element b in G such that a * b = b * a = e, where e is an identity element. In the example, rotating a given angle clockwise and then rotating by the same angle counter-clockwise will leave the square unchanged and the same is true if we reverse the order, i.e. first counter-clockwise and then clockwise. Also, reflecting along a diagonal, say, can be inverted by applying the same reflection again. In symbols:

rot_270° * rot_90°=rot_90° * rot_270° = id and ref_D1 *ref_D1 = id.

History

Main article: Group theory#History

Groups of permutations were already being studied in the 18th century and were applied to solve problems in the theory of equations. However, the formal notion of a group was not published until the late 19th century, and by this time groups had found applications in number theory as well as in geometry.

Basic concepts in group theory

Main article: Glossary of group theory

Order of groups and elements

Main article: Order (group theory)

The order of a group G, usually denoted by |G| or occasionally by o(G), is the number of elements in the set G. If the order is not finite, then the group is an infinite group, denoted |G| = ∞.

The order of an element a in a group G is the least positive integer n such that aⁿ = e, where aⁿ represents Failed to parse (Missing texvc executable; please see math/README to configure.): \underbrace{a * \cdots * a}_n , i.e. application of the operation * to n copies of the value a. (If * represents multiplication, then aⁿ corresponds to the n^th power of a.) If no such n exists, then the order of a is said to be infinity.

In the above example, the order of rot_90° is four, because rotating 4 times by 90° is not changing anything. The order of the reflection elements ref_V etc. is two.

Subgroups

Main article: Subgroup

A set H is a subgroup of a group G if it is a subset of G and a group using the operation defined on G. In other words, H is a subgroup of (G, *) if the restriction of * to H is a group operation on H.

A necessary and sufficient condition for a subset H of a group G to be a subgroup is that g⁻¹h ∈ H for all g, h ∈ H. (See subgroup test.)

If G is a finite group, then so is H. Further, the order of H divides the order of G (Lagrange's Theorem).

The powers of any element a and their inverses (that is, a⁰ = e, a, a², a³, a⁴, …, a⁻¹, a⁻², a⁻³, a⁻⁴, …) always form a subgroup of the larger group. It is said that a generates that subgroup.

A subgroup H always defines a set of left and right cosets. Given an arbitrary element g in G, the left coset of H containing g is

Failed to parse (Missing texvc executable; please see math/README to configure.): gH=\{gh:h\in H\}

and the right coset of H containing g is

Failed to parse (Missing texvc executable; please see math/README to configure.): Hg=\{hg:h\in H\}.

The set of left cosets of H forms a partition of the elements of G; that is, two left cosets are either equal or have an empty intersection. The same holds true of the right cosets of H. In general, the left cosets of H are not necessarily equal to the right cosets of H. If it is the case that for all g in G, gH=Hg, then H is said to be a normal subgroup.

Quotient groups

Main article: Quotient group

If N is a normal subgroup of G, its set of left cosets and right cosets are the same and one may speak simply of the set of cosets of N. In this case, the set of cosets of N may be equipped with an operation (sometimes called coset multiplication, or coset addition) to form a new group, called the quotient group G/N. The operation between the cosets behaves in the nicest way possible: (Ng)·(Nh)=N(gh) for all g and h in G. Note that the coset N itself serves as the identity in this group, and the inverse of Ng in the quotient group is (Ng)Failed to parse (Missing texvc executable; please see math/README to configure.): ^{-1} =N(gFailed to parse (Missing texvc executable; please see math/README to configure.): ^{-1} ).

Simple group

Main article: Simple group

If a group G is not the trivial group and its only normal subgroups are the trivial group and the group itself, it is called a simple group. With the notion of quotient groups, it can be phrased equivalently as: A group with only the trivial group and the group itself as quotient groups is simple.

Group homomorphisms

Main article: Group homomorphism

If G and H are two groups, a group homomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is a mapping Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

GFailed to parse (Missing texvc executable; please see math/README to configure.): \rightarrow

H that preserves the operation between elements. That is, if g and k are any two elements in G, then Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (gk)=Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (g)Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (k). This requirement ensures that Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (1_G)=1_H, and also Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (g)^-1=Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (g^-1) for all g in G.

Two groups G and H are called isomorphic if there exists a group homomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

between G and H which is both surjective (onto) and injective (one-to-one).

The kernel of a homomorphism Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi

is denoted kerFailed to parse (Missing texvc executable; please see math/README to configure.): \varphi
and is the set of elements in G which are mapped to the identity in H. That is, kerFailed to parse (Missing texvc executable; please see math/README to configure.): \varphi

={g in G : Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (g)=1_H}. The kernel of a homomorphism is always a normal subgroup. The First Isomorphism Theorem states that the image of a group homomorphism, Failed to parse (Missing texvc executable; please see math/README to configure.): \varphi (G) is isomorphic to the quotient group G/kerFailed to parse (Missing texvc executable; please see math/README to configure.): \varphi . A useful fact concerning homomorphisms is that they are injective if and only if their kernel is trivial (i.e. kerFailed to parse (Missing texvc executable; please see math/README to configure.): \varphi ={1_G}).

Cyclic groups

Main article: Cyclic group

A cyclic group is a group whose elements may be generated by successive composition of the operation defining the group being applied to a single element of that group. This single element is called the generator or primitive element of the group.

A multiplicative cyclic group in which G is the group, and a is the generator:

Failed to parse (Missing texvc executable; please see math/README to configure.): G = \{ a^n \mid n \in \Z \}

An additive cyclic group, with generator a:

Failed to parse (Missing texvc executable; please see math/README to configure.): G' = \{ n \cdot a \mid n \in \Z \}

If successive composition of the operation defining the group is applied to a non-primitive element of the group, a cyclic subgroup is generated. The order of the cyclic subgroup divides the order of the group. Thus, if the order of a group is prime, all of its elements, except the identity, are primitive elements of the group.

It is important to note that a group contains all of the cyclic subgroups generated by each of the elements in the group. However, a group constructed from cyclic subgroups is itself not necessarily a cyclic group. For example, the Klein four-group is not a cyclic group even though it is constructed from two copies of the cyclic group of order 2.

Abelian groups

Main article: Abelian group

A group Failed to parse (Missing texvc executable; please see math/README to configure.): G

is said to be abelian, or commutative, if the operation satisfies the commutative law. That is, for all Failed to parse (Missing texvc executable; please see math/README to configure.): a
and Failed to parse (Missing texvc executable; please see math/README to configure.): b
in Failed to parse (Missing texvc executable; please see math/README to configure.): G

, Failed to parse (Missing texvc executable; please see math/README to configure.): a*b=b*a . If not, the group is called non-abelian or non-commutative. The name "abelian" comes from the Norwegian mathematician Niels Abel. The above example of symmetries of the square is non-abelian, because

rot_90°*ref_V=ref_D2 ≠ ref_D1=ref_V*rot_90°

The center is group of elements commuting with every other element in the group. The center may range between the whole group and the trivial group.

Notations and remarks

Group operation

Groups can use different notation depending on the context and the group operation.

• Additive groups use + to denote addition, and the minus sign – to denote inverses. For example, a + (–a) = 0 in Z.
• Multiplicative groups use *, Failed to parse (Missing texvc executable; please see math/README to configure.): \cdot

, or the more general 'composition' symbol Failed to parse (Missing texvc executable; please see math/README to configure.): \circ

to denote multiplication, and the superscript –1 to denote inverses. For example, a*a–1 = 1. It is very common to drop the * and just write aa–1 instead.

• Function groups use • to denote function composition, and the superscript ^–1 to denote inverses. For example, g • g^–1 = e. It is very common to drop the • and just write gg^–1 instead.

Omitting a symbol for an operation is generally acceptable, and leaves it to the reader to know the context and the group operation.

When defining groups, it is standard notation to use parentheses in defining the group and its operation. For example, (H, +) denotes the group formed by the set H with addition as group operation. For groups like (Z_n, +) and (F_q^*, *), the multiplicative group of nonzero elements in the finite field F_q, it is common to drop the parentheses and the operation (since only one operation makes these set into a group), as Z_n and F_q^*. It is also correct to refer to a group by its set identifier, e.g. H or Failed to parse (Missing texvc executable; please see math/README to configure.): \Z , or to define the group in set-builder notation, provided it is clear which group operation is intended.

Identity element

Using the identity element property, it can be shown that a group has exactly one identity element. Therefore one usually speaks of the identity: suppose both e and f are identity elements. Then, because f is a (right) identity element e * f = e, and because e is a (left) identity element e * f = f, whence e = f.

The identity element e is sometimes known as the "neutral element," and is sometimes denoted by some other symbol, depending on the group:

• In multiplicative groups, the identity element can be denoted by 1.
• In invertible matrix groups, the identity element is usually denoted by I or Id.
• In additive groups, the identity element may be denoted by 0.
• In function groups, the identity element is usually denoted by f₀.

If S is a subset of G and x an element of G, then, in multiplicative notation, xS is the set of all products {xs : s in S}; similarly the notation Sx = {sx : s in S}; and for two subsets S and T of G, we write ST for {st : s in S, t in T}. In additive notation, we write x + S, S + x, and S + T for the respective sets (see cosets).

Inverse

The inverse of an element a can also be shown to be unique, and it is usually written a⁻¹ or −a, depending on the context. Suppose given an inverse l and another inverse r. Then

l = l * e = l * (a * r) = (l * a) * r = e * r = r.

Moreover, if in a group we know only that b * a = e, then this suffices to conclude that b is the inverse element of a (since a two-sided inverse of a is guaranteed to exist, and then b must be equal to it). Similarly a * b = e suffices for the same conclusion.

(However, a set with a binary operation can have many left identity elements or many right identity elements, provided it has none of the opposite kind: take for instance on any set the operation defined by a * b = b, then any element is a left identity element, but none is a right identity element. Similarly in a monoid an element can have multiple left inverse elements, provided it has no right inverse elements (and vice versa): the set of all maps from an infinite set X to itself is a monoid under function composition, in which every injective map has a left inverse, and every surjective map has a right inverse, but neither of these inverses is unique in general. Yet if all elements in a monoid have a left inverse, the monoid can be shown to be a group.)

Associativity

For a sequence of multiple factors in a given order, one can form a product in many different ways by inserting parentheses; however, by several applications of the associativity property, any two of these can be shown to be equal. For this reason the expression

a₁ * a₂ * ··· * a_n

is unambiguous and parentheses are usually omitted in such expressions. As a consequence it is hardly ever necessary to explicitly invoke the associativity property.

Variants of the definition

Some definitions of a group use seemingly weaker conditions for identity and inverse elements. Instead of requiring a two-sided identity element, one may separately require the existence of a left and right identity element, and similarly one may separately require the existence of a left and right inverse elements: in both cases the left and right elements can be shown to be the same (and each is unique).

Examples of groups

Main articles: Examples of groups and List of small groups

Trivial group

A trivial group is a group consisting of a single element e, with group operation e*e=e.

An abelian group: the integers under addition

A familiar group is the group of integers under addition. Let Z be the set of integers, {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...}, and let the symbol "+" indicate the operation of addition. Then (Z,+) is a group.

Proof:

• Closure: If a and b are integers then a + b is an integer.
• Associativity: If a, b, and c are integers, then (a + b) + c = a + (b + c).
• Identity element: 0 is an integer and for any integer a, 0 + a = a + 0 = a.
• Inverse elements: If a is an integer, then the integer −a satisfies the inverse rules: a + (−a) = (−a) + a = 0.

This group is also abelian because a + b = b + a.

If we extend this example further by considering the integers with both addition and multiplication, it forms a more complicated algebraic structure called a ring. (But, note that the integers with multiplications are not a group.)

An abelian group: the nonzero integers under multiplication modulo a prime

Main article: Multiplicative group of integers modulo n

The nonzero classes of integers modulo p, a prime number, form a group under multiplication. The product of two integers neither of which is divisible by p is not divisible by p either (because p is prime), which shows that the indicated set of classes is closed under multiplication. Associativity is clear, and the class of 1 is the identity for multiplication, so it remains to prove is that each element has an inverse. Fix an integer a not divisible by p. Then multiplication by the class of a is an injective operation, in other words if a * x and a * y are in the same class modulo p, then x and y must be in the same class modulo p (since p divides a * x - a * y = a * (x - y) while it does not divide a). Since our set of classes is finite, and multiplication by a is an injective map from this set to itself, it must be a permutation of these classes, which means it is surjective as well (any nonzero class can be reached from another by multiplication by a). In particular the class of 1 is in the image, say 1 is in the class modulo p of b * a; then the class of b is an inverse for the class of a.

Cyclic multiplicative groups

In the case of a cyclic multiplicative group G, all of the elements aⁿ of the group are generated by the set of all integer exponentiations of a primitive element of that group:

Failed to parse (Missing texvc executable; please see math/README to configure.): G = \{ a^n \mid n \in \Z \pmod{m \in \Z} \}

In this example if a is 2 and the operation is the mathematical multiplication operator, then

Failed to parse (Missing texvc executable; please see math/README to configure.): G = \{ .., 2^{-2}, 2^{-1}, 2^0, 2^1, 2^2, 2^3, .. \} = \{ .., 0.25, 0.5, 1, 2, 4, 8, .. \}.\,

The modulo m may bind the group into a finite set with a non-fractional set of elements, since the inverse (and Failed to parse (Missing texvc executable; please see math/README to configure.): x^{-2}

, etc.) would be within the set.

Not a group: the integers under multiplication

On the other hand, if we consider the integers with the operation of multiplication, denoted by "·", then (Z,·) is not a group. It satisfies most of the axioms, but fails to have inverses:

• Closure: If a and b are integers then a · b is an integer.
• Associativity: If a, b, and c are integers, then (a · b) · c = a · (b · c).
• Identity element: 1 is an integer and for any integer a, 1 · a = a · 1 = a.
• However, it is not true that whenever a is an integer, there is an integer b such that ab = ba = 1. For example, a = 2 is an integer, but the only solution to the equation ab = 1 in this case is b = 1/2. We cannot choose b = 1/2 because 1/2 is not an integer. (Inverse element fails)

Since not every element of (Z,·) has an inverse, (Z,·) is not a group. It is, however, a commutative monoid, which is a similar structure to a group but does not require inverse elements.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, the set of all fractions of integers a/b, where a and b are integers and b is nonzero, and the operation multiplication, denoted by "·". Since the rational number 0 does not have a multiplicative inverse, (Q,·), like (Z,·), is not a group.

However, if we instead use the set of all nonzero rational numbers Q \ {0}, then (Q \ {0},·) does form an abelian group.

• Closure, Associativity, and Identity element axioms are easy to check and follow because of the properties of integers.
• Inverse elements: The inverse of a/b is b/a and it satisfies the axiom.

We don't lose closure by removing zero, because the product of two nonzero rationals is never zero. Just as the integers form a ring, the rational numbers form the algebraic structure of a field, allowing the operations of addition, subtraction, multiplication and division.

A finite nonabelian group: permutations of a set

This example is taken from the larger article on the Dihedral group of order 6

For a more concrete example of a group, consider three colored blocks (red, green, and blue), initially placed in the order RGB. Let a be the action "swap the first block and the second block", and let b be the action "swap the second block and the third block".

In multiplicative form, we traditionally write xy for the combined action "first do y, then do x"; so that ab is the action RGB → RBG → BRG, i.e., "take the last block and move it to the front". If we write e for "leave the blocks as they are" (the identity action), then we can write the six permutations of the set of three blocks as the following actions:

• e : RGB → RGB
• a : RGB → GRB
• b : RGB → RBG
• ab : RGB → BRG
• ba : RGB → GBR
• aba : RGB → BGR

Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,

• bb = e,
• (aba)(aba) = e, and
• (ab)(ba) = (ba)(ab) = e;

so each of the above actions has an inverse.

By inspection, we can also determine associativity and closure; note for example that

• (ab)a = a(ba) = aba, and
• (ba)b = b(ab) = bab.

This group is called the symmetric group on 3 letters, or S₃. It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab ≠ ba). Since S₃ is built up from the basic actions a and b, we say that the set {a,b} generates it.

More generally, we can define a symmetric group from all the permutations of N objects. This group is denoted by S_N and has order N factorial.

One of the reasons that permutation groups are important is that every finite group can be expressed as a subgroup of a symmetric group S_N; this result is Cayley's theorem.

Elementary group theory

Elementary group theory is concerned with basic facts that hold for all individual groups. For example:

• You can perform division in groups; that is, given elements a and b of the group G, there is exactly one solution x in G to the equation x * a = b and exactly one solution y in G to the equation a * y = b. In fact, right respectively left multiplication of the equation by a^-1 gives the solution x = b * a^-1 respectively y = a^-1 * b.

• (Socks and shoes) The inverse of a product is the product of the inverses in the opposite order: (a * b)⁻¹ = b⁻¹ * a⁻¹.

Proof: We will demonstrate that (a * b) * (b^-1 * a^-1) = e, which as mentioned above suffices to prove that b^-1 * a^-1 is the inverse of a * b.

Failed to parse (Missing texvc executable; please see math/README to configure.): (a*b)*(b^{-1}*a^{-1})	=	Failed to parse (Missing texvc executable; please see math/README to configure.): (a*(b*b^{-1}))*a^{-1}	(associativity)
	=	Failed to parse (Missing texvc executable; please see math/README to configure.): (a*e)*a^{-1}	(definition of inverse)
	=	Failed to parse (Missing texvc executable; please see math/README to configure.): a*a^{-1}	(definition of neutral element)
	=	Failed to parse (Missing texvc executable; please see math/README to configure.): e	(definition of inverse)

Constructing new groups from given ones

Some possible ways to construct new groups from a set of given groups:

• Subgroups: The closure, under the group operation and inversion, of any nonempty subset of a group is also a group.
• Quotient group: Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation (gN)(hN)=ghN.
• Direct product: If (G,*) and (H,•) are groups, then the set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁*g₂,h₁•h₂) is a group. The direct product can also be defined with any number of terms, finite or infinite, by using the Cartesian product and defining the operation coordinate-wise.
• Semidirect product: If N and H are groups and φ : H → Aut(N) is a group homomorphism, then the semidirect product of N and H with respect to φ is the group (N × H, *), with * defined as

  (n₁, h₁) * (n₂, h₂) = (n₁ φ(h₁) (n₂), h₁ h₂)

• Direct external sum: The direct external sum of a family of groups is the subgroup of the product constituted by elements that have a finite number of non-identity coordinates. If the family is finite the direct sum and the product are equivalent.

Groups with additional structure

In differential geometry, algebraic geometry, and topology, the group concept specializes to include groups with additional structure. Lie groups, algebraic groups and topological groups are examples of group objects: group-like structures sitting in a category other than the ordinary category of sets.

Generalizations

In abstract algebra, more general structures arise by relaxing some of the axioms defining a group.

• Eliminating the requirement that every element have an inverse, then the resulting algebraic structure is called a monoid.
• A monoid without an identity is called a semigroup.
• Alternatively, relaxing the requirement that the operation be associative while still requiring the possibility of division, the resulting algebraic structure is a loop.
• A loop without an identity is called a quasigroup.
• Finally, dropping all axioms for the binary relation, the resulting algebraic structure is called a magma.

Groupoids, which are similar to groups except that the composition a * b need not be defined for all a and b, arise in the study of more involved kinds of symmetries, often in topological and analytical structures. Groupoids, in turn, are special sorts of categories.

Supergroups and Hopf algebras are other generalizations.

Abelian groups form the prototype for the concept of an abelian category, which has applications to vector spaces and beyond.

Formal group laws are certain formal power series which have properties much like a group operation.

References

• Herstein, I. N. (1996), Abstract algebra (3 ed.), Upper Saddle River, NJ: Prentice Hall Inc., MR1375019, ISBN 978-0-13-374562-7
• Dummit, David S. & Foote, Richard M. (2004), Abstract algebra (3 ed.), New York: Wiley, MR2286236, ISBN 978-0-471-43334-7
• Lang, Serge (2002), Algebra, vol. 211, Graduate Texts in Mathematics, Berlin, New York, MR1878556, ISBN 978-0-387-95385-4

See also

Wikibooks has a book on the topic of

Abstract algebra/Groups

External links

• Eric W. Weisstein, Group at MathWorld.
• Group at PlanetMath.af:Groep (wiskunde)

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