判定

  1. 封闭性
  2. 结合律
  3. 存在单位元 e
  4. 对任意元素存在逆元

Properties

  • Let G be a group and a and b be any two elements in G. Then the equations and have unique solutions in G.
  • There is a unique e (identity element) => , its order is 1, the inverses in a group are also unique
  • The right and left cancellation laws are true in groups

Definition

Group theory - Wikiwand
In mathematics, a group is a set and an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse.

  • abelian or commutative
    A group G with the property that a ◦ b = b ◦ a for all a, b ∈ G is called abelian or commutative.

  • the general linear group
    The set of invertible matrices forms a group called the general linear group.

  • Order

    • The order of a finite group is the number of elements it contains.
    • The order of a element is the minimum positive number such that , . If there is no such number, then has infinite order.

Notation

If the group is or , we write the group operation additively and the exponential operation multiplicatively; that is, we write instead of .

Groups are essentially a Set

Subgroups