Group (mathematics)

In mathematics, a group is a set endowed with a binary operation satisfying certain axioms, detailed below. For example, the set of integers with addition as the binary operation is a group. Group theory is the branch of mathematics which studies groups.

Group theory originated with the work of Évariste Galois, in 1830, on the problem of when an algebraic equation is soluble by radicals. Before this work, groups were mainly studied in terms of permutations. Some aspects of abelian group theory were also known in the theory of quadratic forms.

Many of the structures investigated 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 one to study such structures in a general setting.

Group theory has extensive applications in mathematics, science, and engineering. Many algebraic structures such as fields and vector spaces are based on groups, and group theory provides an important tool for studying symmetry, since the symmetries of any object form a group. Groups are thus pertinent to branches of sciences involving symmetry principles, such as relativity, quantum mechanics, particle physics, chemistry, computer graphics, and others.

History

See Group theory.

Basic definitions

A group (G, * ) is a set G along with a function * : G × G → G, satisfying the group axioms below. The function * is called a binary operator. Here "a * b" represents the result of applying the function * to the ordered pair (a, b) of elements in G. The group axioms are the following:

• Associativity: For every a, b and c in G, (a * b) * c = a * (b * c).
• Neutral element: There is an element e in G such that for every a in G, e * a = a * e = a.
• Inverse element: For every a in G, there is an element b in G such that a * b = b * a = e, where e is the neutral element from the previous axiom.

One often also sees the axiom:

• Closure: For all a and b in G, a * b belongs to G.

Since the definition of group given here uses the notion of binary operation, closure is automatically satisfied and hence would be superfluous as an axiom. When determining if a given * is a group operation, one nevertheless checks that * satisfies closure as part of verifying that it is, in fact, a binary operation.

The neutral element of a group is often called the identity element if the operation is written in multiplicative notation, while it is called the zero element or null element if the operation is written in additive notation.

If a group has both a left neutral element (say, e₁) and a right neutral element (say, e₂), then they must be identical (because e₁ = e₁ * e₂ = e₂). It follows that the neutral element e of the second group axiom is unique, that is, a group has only one neutral element. This is why the third group axiom refers to the neutral element, even though the second axiom merely asserts that there is at least one neutral element.

The order of a group G, denoted by |G| or o(G), is the number of elements of the set G. A group is called finite if it has finitely many elements, that is if the set G is a finite set.

When there is no ambiguity, the group (G, * ) is often denoted simply as "G", leaving the operation * unmentioned. But different operations on the same set would define different groups.

The operation in a group need not be commutative, that is there may exist elements a,b such that a * b ≠ b * a. A group G is said to be abelian (after the mathematician Niels Abel) (or commutative) if for every a, b in G, a * b = b * a. Groups lacking this property are called non-abelian.

Alternative axiomatizations

The axioms given above in the definition of group are stronger than what is strictly required. Sufficient are associativity, the existence of a right neutral element (that is, there is an element e such that x * e = x for all x), and the existence of right inverses with respect to this right neutral element (that is, for each x, there is a y such that x * y = e). It follows from these that the postulated right neutral element e is also a left neutral element, and hence, as above, is unique. Further, it follows that each right inverse is also a left inverse. Thus, the axiomatization given above is not strictly minimal in the logical sense; however, it is customary. One reason for the custom is that the axioms as given are easily remembered and checked in practice. Another reason is that subsets or variants of the axioms define other useful algebraic structures — e.g., groupoids and semigroups.

Groups can be axiomatized in ways other than the one presented above. For instance, a group is a set G closed under:

• An associative binary operation, here denoted by concatenation, and a unary operation, denoted by the superscript -1 such that x^-1xy = yx^-1x = y is an axiom. It follows that x^-1x is a constant, and hence, is the neutral element.
• An associative binary operation, here denoted by concatenation, such that for each a,b ∈ G, there exist x,y ∈ G such that ax = b and ya = b. Equivalently, a group is an associative quasigroup. The existence of the neutral element follows easily.
• Two binary operations, here denoted by infix "/" and "\", with axioms y = x/(y\x) = (x/y)\x, and (x/y)\z = x/(y\z). One may then define an associative binary operation and a unary operation as above in terms of \ and / so that x/y = xy^-1 and x\y = x^-1y.

Notation for groups

Convention	Multiplication	Addition
Operation	x * y or xy	x + y
Identity	e or 1	0
Powers	xn	nx
Inverse	x−1	−x
Direct sum	G × H	G ⊕ H

Usually the operation, whatever it really is, is thought of as an analogue of multiplication, and the group operations are therefore written multiplicatively. That is:

• We write "a · b" or even "ab" for a * b and call it the product of a and b;
• We write "1" (or "e") for the neutral element and call it the unit element;
• We write "a⁻¹" for the inverse of a and call it the reciprocal of a.

However, sometimes the group operation is thought of as analogous to addition and written additively:

• We write "a + b" for a * b and call it the sum of a and b;
• We write "0" for the neutral element and call it the zero element;
• We write "−a" for the inverse of a and call it the opposite of a.

Usually, only abelian (commutative) groups are written additively, although they may also be written multiplicatively. When being noncommittal, one can use the notation (with "*") and terminology that was introduced in the definition, using the notation a⁻¹ for the inverse of a.

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.

Some elementary examples and nonexamples

An abelian group: the integers under addition

A group that we are introduced to in elementary school is the integers under addition. For this example, 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 (written additively).

Proof:

• If a and b are integers then a + b is an integer. (Closure; + really is a binary operation)
• If a, b, and c are integers, then (a + b) + c = a + (b + c). (Associativity)
• 0 is an integer and for any integer a, 0 + a = a + 0 = a. (Identity element)
• If a is an integer, then there is an integer b := −a, such that a + b = b + a = 0. (Inverse element)

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

The integers with both addition and multiplication together form the more complicated algebraic structure of a ring. In fact, the elements of any ring form an abelian group under addition, called the additive group of the ring.

Not a group: the integers under multiplication

On the other hand, if we consider the operation of multiplication, denoted by "·", then (Z,·) is not a group:

• If a and b are integers then a · b is an integer. (Closure)
• If a, b, and c are integers, then (a · b) · c = a · (b · c). (Associativity)
• 1 is an integer and for any integer a, 1 · a = a · 1 = a. (Identity element)
• 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. The most we can say is that it is a commutative monoid.

An abelian group: the nonzero rational numbers under multiplication

Consider the set of rational numbers Q, that is the set of numbers a/b such that 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 Q \ {0} instead of Q, that is include every rational number except zero, then (Q \ {0},·) does form an abelian group (written multiplicatively). The inverse of a/b is b/a, and the other group axioms are simple to check. 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. In fact, the nonzero elements of any given field form a group under multiplication, called the multiplicative group of the field.

A finite nonabelian group: permutations of a set

For a more concrete example, 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".

Cycle diagram for S₃. A loop specifies a series of powers of any element connected to the identity element (1). For example, the e-ba-ab loop reflects the fact that (ba)²=ab and (ba)³=e, as well as the fact that (ab)²=ba and (ab)³=e The other "loops" are roots of unity so that, for example a²=e.

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.

Every group can be expressed in terms of permutation groups like S₃; this result is Cayley's theorem and is studied as part of the subject of group actions.

Further examples

For some further examples of groups from a variety of applications, see Examples of groups and List of small groups.

Simple theorems

• A group has exactly one identity element.

• Every element has exactly one inverse.

Proof: Suppose both b and c are inverses of x. Then, by the definition of an inverse, xb = bx = e and xc = cx = e. But then:

$xb=e=xc$
$xb=xc$
$bxb=bxc$	(multiplying on the left by b)
$eb=ec$	(using bx = e)
$b=c$	(neutral element axiom)

Therefore the inverse is unique.

The first two properties actually follow from associative binary operations defined on a set. Given a binary operation on a set, there is at most one identity and at most one inverse for any element.

• 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.

• The expression "a₁ * a₂ * ··· * a_n" is unambiguous, because the result will be the same no matter where we place parentheses.

• (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 (ab)(b^-1a^-1) = (b^-1a^-1)(ab) = e, as required by the definition of an inverse.

$(ab)(b^{-1}a^{-1})$	=	$a(bb^{-1})a^{-1}$	(associativity)
	=	$aea^{-1}$	(definition of inverse)
	=	$aa^{-1}$	(definition of neutral element)
	=	$e$	(definition of inverse)

And similarly for the other direction.

These and other basic facts that hold for all individual groups form the field of elementary group theory.

Constructing new groups from given ones

1. If a subset H of a group (G,*) together with the operation * restricted on H is itself a group, then it is called a subgroup of (G,*).
2. The direct product of two groups (G,*) and (H,•) is the Cartesian product set G×H together with the operation (g₁,h₁)(g₂,h₂) = (g₁*g₂,h₁•h₂). 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.
3. The semidirect product of two groups N and H with respect to a group homomorphism φ : H → Aut(N) is a new group (N × H, *), with * defined as

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

4. 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 of course the same.
5. 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.

Generalizations

In abstract algebra, we get some related structures which are similar to groups by relaxing some of the axioms given at the top of the article.

• If we eliminate the requirement that every element have an inverse, then we get a monoid.
• If we additionally do not require an identity either, then we get a semigroup.
• Alternatively, if we relax the requirement that the operation be associative while still requiring the possibility of division, then we get a loop.
• If we additionally do not require an identity, then we get a quasigroup.
• If we don't require any axioms of the binary operation at all, then we get 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. They are special sorts of categories.

Supergroups and Hopf algebras are other generalizations.

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.

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.

Attribution

Some content on this page may previously have appeared on Wikipedia.

References

• Herstein, I.N. Abstract Algebra, Wiley, ISBN 0-471-36879-2
• Dummit, David and Richard Foote. Abstract Algebra, Wiley, ISBN 0-471-43334-9
• Lang, Serge. Algebra, Springer, ISBN 0-387-95385-X