Encyclopedia
In
mathematics, a
group is a
set together with a binary operation satisfying certain axioms, detailed below. For example, the set of integers is a group when addition is taken as the binary operation. The branch of mathematics which studies groups is called group theory.
Group theory originated with the work of Évariste Galois in 1830, which concerned the problem of when an
algebraic equation is soluble by radicals. Prior to 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 for the properties of such structures to be investigated in a general setting.
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.
History
See Group theory.
Basic definitions
A group is a
set G together with a binary operation * :
G ×
G ?
G, satisfying the group axioms below. "
a *
b" represents the result of applying the operation * to the ordered pair of elements of
G. The group axioms are the following:
- Associativity: For all a, b and c in G, * c = a * .
- Neutral element: There is 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 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 and a right neutral element , then they must be identical . 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, 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.
The group is often referred to as simply "
G", leaving the operation * unmentioned. But to be perfectly precise, different operations on the same set define different groups.
The operation in a group need not be commutative, that is there may exist elements such that
a *
b ?
b *
a. A group
G is said to be
abelian 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 , and the existence of right inverses with respect to this right neutral element . 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/ = \x, and \z = x/. 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" 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 groups are written additively, although they may also be written multiplicatively. When being noncommittal, one can use the notation 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 ; similarly the notation
Sx = ; and for two subsets
S and
T of
G, we write
ST for . 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, , and let the symbol "+" indicate the operation of addition.
Then is a group .
Proof:
- If a and b are integers then a + b is an integer.
- If a, b, and c are integers, then + c = a + .
- 0 is an integer and for any integer a, 0 + a = a + 0 = a.
- If a is an integer, then there is an integer b := -a, such that a + b = b + a = 0.
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 is not a group:
- If a and b are integers then a · b is an integer.
- If a, b, and c are integers, then · c = a · .
- 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.
Since not every element of has an inverse, 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, , like , is not a group.
However, if we instead use the set
Q \ instead of
Q, that is include every rational number
except zero, then
does form an abelian group .
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 , 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" , 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,
so each of the above actions has an inverse.
By inspection, we can also determine associativity and closure; note for example that
- a = a = aba, and
- b = b = bab.
This group is called the
symmetric group on 3 letters, or
S3.
It has order 6 , and is non-abelian .
Since
S3 is built up from the basic actions
a and
b, we say that the set
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:
- 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 * ··· * an" is unambiguous, because the result will be the same no matter where we place parentheses.
- The inverse of a product is the product of the inverses in the opposite order: -1 = b-1 * a-1.
- Proof: We will demonstrate that = = e, as required by the definition of an 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
- If a subset H of a group together with the operation * restricted on H is itself a group, then it is called a subgroup of .
- The direct product of two groups and is the set G×H together with the operation = . 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.
- The semidirect product of two groups N and H with respect to a group homomorphism φ : H → Aut is a new group , with * defined as
- * =
- 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.
- Given a group G and a normal subgroup N, the quotient group is the set of cosets of G/N together with the operation =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.
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
See also
- Glossary of group theory
- Elementary group theory
- List of group theory topics
- Important publications in group theory
- Examples of groups
...
External links
