# group (algebra)

## 1. definition: group

A non-empty set \(G\) is a group if it has a defined operation \((\cdot)\) such that:

- (closure) \(a,b\in G \Rightarrow a\cdot b \in G\)
- (associative law) If \(a,b,c\in G\), then \(a\cdot (b \cdot c) = (a \cdot b) \cdot c\)
- (identity element) There exists an \(e\in G\) such that \(a \cdot e = e \cdot a = a\) for all \(a \in G\)
- (inverse) For every \(a\in G\), there exists \(a^{-1}\in G\) such that \(a\cdot a^{-1} = a^{-1}\cdot a = e\)

## 2. related algebraic structures

#### 2.0.1. abelian group

A group where \(a\cdot b = b\cdot a\)

#### 2.0.2. definition: semi-group

A group that does not necessarily have an identity element or an inverse for every element.

#### 2.0.3. definition: monoid

A group that does not necessarily have an inverse for every element. Or, a semigroup with an identity element.

## 3. sources

- Topics in Algebra – Herstein 3rd edition
- wikipedia entry on algebraic structures