Group Definition

If we have element a and operator * that conform the following property.

group property

1. 封闭性（Closure）

$$
\text{if } a, b \in G \text{ then } a*b \in G
$$

2. 结合律（Associativity）

$$
\text{if } a, b, c \in G \text{ then } (ab)c = a(bc)
$$

3. 同一/恒等/单位元 （Identity）

$$
\exists e \Rightarrow \forall a, e * a = a * e = a
$$

4. 逆元（Inverse）

$$
\text{if } a \in G, \exists a^{-1}, \text{then } a * a^{-1} = e
$$

different group

• |C | A | ID | IN
  –|–|–|–|–
  semigroup| $\checkmark$ | $\checkmark$
  monoid | $\checkmark$ | $\checkmark$ | $\checkmark$
  group | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$