Group Definition

A group is a set of elements $G = \{e, a, b, \dots\}$ and an operation, called group multiplication ($a \cdot b$), which obeys the four properties:

1. Closure $\forall a,b \in G, a \cdot b \in G$
2. Identity $\exists e \in G$ such that $e\cdot a = a\cdot e = a$ $\forall a \in G$
3. Inverse $\forall a \in G$, $\exists a^{-1} \in G$ such that $aa^{-1} = a^{-1}a = e$
4. Associativity $\forall a,b,c \in G$, $a\cdot(b\cdot c) = (a\cdot b) \cdot c$

A group can be abstract or concrete.

• Abstract: the elements do not represent anything, such as in the definition above.
• Concrete: the elements represent mathematical or physical objects, such matrices, numbers, particles, etc.

Consider the multiplication of elements $a,b \in G$.

• If $a\cdot b = b\cdot a$ $\forall a,b \in G$, the group is abelian (or commutative).
• If $a\cdot b \neq a\cdot b$ for at least some $a,b \in G$, the group is non-abelian (or non-commutative).