# Unique things in Group

## 1 Identity in group is unique

### 1.1 Proposition

If $$h \in G$$ is an identity of group $$G$$, then $$h = e_G$$

### 1.2 Proof

$h = h \cdot e_G = e_G$

1. forall $$a$$, holds $$a = a \cdot e_G$$
2. forall $$a$$, holds $$a = h \cdot a$$($$h$$ is an identity).

## 2 Inverse of an element is unique

### 2.1 Proposition

Let $$f$$ has inverses $$a$$ and $$b$$, then $$a = b$$

### 2.2 Proof

$f \cdot a = e = f \cdot b$

By definition of inverse, $$f \cdot f^{-1} = e$$, hence

1. $$f \cdot a = e$$
2. $$f \cdot b = e$$

Then cancel $$f$$, only $$a = b$$ make sense

Date: 2022-11-23 Wed 00:00