Group

A group is a set $G$ together with a binary operation $\cdot :G\times G\to G$ that satisfies the following properties:

1. Closure: For all $a,b\in G$, $a\cdot b\in G$.
2. Associativity: For all $a,b,c\in G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
3. Identity: There exists an element $e\in G$ such that for all $a\in G$, $e\cdot a=a\cdot e=a$.
4. Inverse: For each $a\in G$, there exists an element $a^{-1}\in G$ such that $a\cdot a^{-1}=a^{-1}\cdot a=e$.

Definition (Abelian Group)

An Abelian group is a group $G$ that also satisfies the commutativity property:

$$a\cdot b=b\cdot a\forall a,b\in G.$$

Definition (Free Group)

A group $G$ is called free if it has a basis, which is a subset $B\subseteq G$ such that every element of $G$ can be uniquely expressed as a finite product of elements of $B$ and their inverses.

Examples

• Euclidean Group $E(n)$: The group of isometries of ${\mathbb{R}}^n$.
• Special Euclidean Group $\mathrm{SE}(n)$: The group of orientation-preserving isometries of ${\mathbb{R}}^n$.
• Orthogonal Group $O(n)$: The group of $n\times n$ orthogonal matrices.
• Special Orthogonal Group $\mathrm{SO}(n)$: The group of $n\times n$ orthogonal matrices with determinant $1$.