Basics of groups群ぐん

A group群ぐん is a structure in which operations can be composed and, when necessary, undone. Addition of integers, addition of residue classes, rotations of figures, and products of invertible matrices look different. However, each has a binary operation, which includes closure, together with associativity, an identity, and inverses.

1Definition of a group

A pair $$(G,\text{[PARSE ERROR: Undefined("Command("ast")")]})$$(G,\ast) consisting of a set $$G$$G and a binary operation $$\text{[PARSE ERROR: Undefined("Command("ast")")]}:G\times G\to G$$\ast:G\times G\to G is a group if it satisfies the following conditions. Closure is already included in the statement that $$\text{[PARSE ERROR: Undefined("Command("ast")")]}$$\ast is a binary operation on $$G$$G.

For this same $$e$$e,

The first equality is the associative law for every $$a,b,c\in G$$a,b,c\in G; the next two statements assert the existence of an identity and of an inverse for every element relative to that same identity.

2Why these conditions?

Closure guarantees that the result of an operation stays in the same world. Associativity guarantees that when operations are repeated, the position of parentheses does not matter. The identity element単位元たんいげん is the operation that does nothing, and an inverse is an operation that cancels another operation.

With these conditions, the equation

has the unique solution

Indeed, operating on the left by $$a^{-1}$$a^{-1} gives

so $$x=a^{-1}\text{[PARSE ERROR: Undefined("Command("ast")")]}b$$x=a^{-1}\ast b. Thus every solution is forced to have this value and is unique. Conversely, substituting the value gives $$a\text{[PARSE ERROR: Undefined("Command("ast")")]}\left(a^{-1}\text{[PARSE ERROR: Undefined("Command("ast")")]}b\right)=\left(a\text{[PARSE ERROR: Undefined("Command("ast")")]}{a}^{-1}\right)\text{[PARSE ERROR: Undefined("Command("ast")")]}b=e\text{[PARSE ERROR: Undefined("Command("ast")")]}b=b$$a\ast(a^{-1}\ast b)=(a\ast a^{-1})\ast b=e\ast b=b, so it really is a solution. In a noncommutative group, the side matters: the solution of $$x\text{[PARSE ERROR: Undefined("Command("ast")")]}a=b$$x\ast a=b is $$x=b\text{[PARSE ERROR: Undefined("Command("ast")")]}{a}^{-1}$$x=b\ast a^{-1}.

3Abelian groups

A group群ぐん in which

holds for all $$a,b\in G$$a,b\in G is called an abelian group可換群かかんぐん. Addition of integers is commutative. In contrast, composition of permutations and matrix multiplication are generally not commutative.

Here a permutation is used only as an operation that rearranges elements of a set, and matrix multiplication is used only as an example of a noncommutative operation.

4Basic examples

Here $$R^{\times }$$\mathbb R^\times is the set of all nonzero real numbers. The element 0 is excluded because it has no multiplicative inverse.

The group $$S_n$$S_n is the set of all bijections from $$\{1,\text{[PARSE ERROR: Undefined("Command("dots")")]},n\}$$\{1,\dots,n\} to itself, that is, all permutations. The group $$G{L}_n\left(\mathbb{R}\right)$$GL_n(\mathbb R) is the set of all $$n\times n$$n\times n real matrices that have inverses under multiplication. Detailed matrix calculations are not used as a prerequisite for the proofs on this page.

In $$S_n$$S_n, a composition of bijections is a bijection, composition of maps is associative, the identity map is the identity element, and every bijection has an inverse map. In $$G{L}_n\left(\mathbb{R}\right)$$GL_n(\mathbb R), if $$A$$A and $$B$$B are invertible, then $$(AB{)}^{-1}=B^{-1}{A}^{-1}$$(AB)^{-1}=B^{-1}A^{-1}, so the product remains invertible. Associativity of matrix multiplication, the identity matrix, and inverse matrices provide the other group conditions.

5What changes and what is preserved

When objects are viewed as groups, the concrete appearance of the elements becomes less important. Integers, permutations, and matrices are different kinds of objects. However, the ability to repeat the operation, the existence of an identity element単位元たんいげん, and the ability to return by an inverse are shared. Group theory studies arguments while preserving this common structure.

6Proof supplement: identity elements, inverses, and cancellation

In a group群ぐん, the identity element単位元たんいげん is unique. Suppose $$e$$e and $$e^{\prime }$$e' are both identity elements. Since $$e$$e is an identity element単位元たんいげん, $$e{e}^{\prime }=e^{\prime }$$ee'=e'. Since $$e^{\prime }$$e' is an identity element単位元たんいげん, $$e{e}^{\prime }=e$$ee'=e. Therefore $$e=e^{\prime }$$e=e'.

Inverses are also unique. Suppose $$b$$b and $$c$$c are both inverses of $$a$$a. That means $$ab=e$$ab=e and $$ca=e$$ca=e. Then

Here the associative law結合法則けつごうほうそく is used.

Furthermore, the cancellation law消去法則しょうきょほうそく holds in a group:

and

For the first implication, multiplying by $$a^{-1}$$a^{-1} from the left gives

By associativity, $$\left(a^{-1}a\right)b=\left(a^{-1}a\right)c$$(a^{-1}a)b=(a^{-1}a)c, so $$eb=ec$$eb=ec. Therefore $$b=c$$b=c. The second implication follows similarly by multiplying by $$a^{-1}$$a^{-1} on the right.

This proof証明しょうめい shows that in a group, the ability to undo operations is what gives us the power to solve equations.

7Exercise link

8Summary

A group群ぐん is an algebraic structure代数的構造だいすうてきこうぞう with an associative binary operation, an identity element単位元たんいげん, and an inverse for every element. The reason to study groups is to handle numbers, permutations, matrices, and symmetries in the same language.