Group Definition

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

group property

  1. 封闭性(Closure)

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

  1. 结合律(Associativity)

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

  1. 同一/恒等/单位元 (Identity)

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

  1. 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 | $\checkmark$ | $\checkmark$ | $\checkmark$ | $\checkmark$

Group Definition
https://rugal.github.com/2023/2023-01-31-group-definition/
Author
Rugal Bernstein
Posted on
January 31, 2023
Licensed under