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In abstract algebra, a group is a set with a binary operation that satisfies certain axioms, detailed below. For example, the set of integers with addition is a group. The branch of mathematics which studies groups is called group theory.
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.
A group (G, *) is a set G with a binary operation * that satisfies the following four axioms:
Some texts omit the explicit requirement of closure, since the closure of the group follows from the fact that the operation * is a binary operation.
Using the identity element property it can be shown that a group has exactly one identity element. See Simple theorems.
The inverse of an element can also be shown to be unique, and the left- and right-inverses of an element are the same. Some definitions are thus slightly more narrow, substituting the second and third axioms with the concept of a "left (or right) identity element" and a "left (or right) inverse element."
Also note that a group (G,*) is often denoted simply G where there is no ambiguity in what the operation is.
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 a<sup>n</sup> = e, where a<sup>n</sup> is multiplication of a by itself n times (or other suitable composition depending on the group operator). If no such n exists, then the order of a is said to be infinity.
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.
If G is a finite group, then so is H. Further, the order of H divides the order of G (Lagrange's Theorem).
A group G is said to be an abelian group (or commutative) if the operation is commutative, that is, for all a, b in G, a * b = b * a. A non-abelian group is a group that is not abelian. The term "abelian" is named after the mathematician Niels Abel.
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:
An additive cyclic group, with generator a:
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, a Klein group is not a cyclic group even though it is constructed from two copies of the cyclic group of order 2.
Groups can use different notation depending on the context and the group operation.
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 that the set H is a group under addition. For groups like (Z<sub>n</sub>, +) and (F<sub>n</sub>*, *), it is common to drop the parentheses and the operation, e.g. Z<sub>n</sup> and F<sub>n</sub>*. It is also correct to refer to a group by its set identifier, e.g. H or , or to define the group in set-builder notation.
The identity element e is sometimes known as the "neutral element," and is sometimes denoted by some other symbol, depending on the group:
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).
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:
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, which forms a more complicated algebraic structure called a ring. (But, note that the integers with multiplications are not a group)
In the case of a cyclic multiplicative group G, all of the elements a<sup>n</sup> of the group are generated by the set of all integer exponentiations of a primitive element of that group:
In this example if a is 2 and the operation is the mathematical multiplication operator, then G = = . The modulo m may bind the group into a finite set with a non-fractional set of elements, since the inverse (and , etc.) would be within the set.
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:
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.
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 (number) 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.
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.
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:
Note that the action aa has the effect RGB → GRB → RGB, leaving the blocks as they were; so we can write aa = e. Similarly,
By inspection, we can also determine associativity and closure; note for example that
This group is called the symmetric group on 3 letters, or S<sub>3</sub>. It has order 6 (or 3 factorial), and is non-abelian (since, for example, ab ≠ba). Since S<sub>3</sub> 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<sub>N</sub> 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<sub>N</sub>; this result is Cayley's theorem.
Proof: Suppose both e and f are identity elements. Then, by the definition of identity, fe = ef = e and also ef = fe = f. But then e = f.
Therefore the identity element is unique.
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.
Proof: We will demonstrate that (ab)(b<sup>-1</sup>a<sup>-1</sup>) = (b<sup>-1</sup>a<sup>-1</sup>)(ab) = 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.
Some possible ways to construct new groups from a set of given groups:
There are two main methods in proving that a set is a group:
The first method is generally referred to as the "Subgroup Test" and requires that you prove the following if trying to prove that H is a subgroup:
The second method requires that you prove all the axioms and assumptions in the definition for a set G:
For finite groups, one only needs to prove that a subset is non-empty and is closed under the ambient group's operation.
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.
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.
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