# 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

\[ h = h \cdot e_G = e_G \]

- forall \(a\), holds \(a = a \cdot e_G\)
- 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

- \(f \cdot a = e\)
- \(f \cdot b = e\)

Then cancel \(f\), only \(a = b\) make sense