Group

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 $ax=b$ and $xa=b$ have unique solutions in G.
• There is a unique e (identity element) ⇒ $e\cdot a=a\cdot e=a$ , 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 $g\in G$ is the minimum positive number $m$ such that $g^m=e$, $ord(g)=m$. If there is no such number, then $g$ has infinite order.

Notation

If the group is $\mathbb{Z}$ or ${\mathbb{Z}}_n$ , we write the group operation additively and the exponential operation multiplicatively; that is, we write $ng$ instead of $g^n$.

Groups are essentially a Set

Subgroups