Non-Abelian Groups

A non-abelian group, or a non-commutative group, refers to a category of groups where the binary operation does not follow the commutative law.

Simply put, for at least two elements $a$ and $b$ in the group, the equation $a * b \neq b * a$ holds, with $ * $ symbolizing the group's binary operation.

To be recognized as a group $ (G, *) $, it must fulfill these essential criteria:

Closure
For any two elements $a, b \in G$, the result of $a * b$ remains within $G$.
Associativity
For any $a, b, c \in G$, the operation follows the rule $(a * b) * c = a * (b * c)$.
Identity Element
An element $e \in G$ exists such that for any $a \in G$, it's true that $a * e = e * a = a$.
Inverse Element
For every $a \in G$, an element $b \in G$ can be found so that $a * b = b * a = e$, with $e$ being the identity element of the group.

However, what sets a group apart as non-abelian or non-commutative is the absence of commutativity for at least a pair of elements $a, b \in G$, meaning $a * b \neq b * a$.

This detail underlines that the sequence in which elements are combined directly impacts the operation's outcome.

Non-abelian groups play a significant role across various mathematical and physical disciplines, such as group theory, algebra, and quantum mechanics, where the understanding of symmetries and the essence of non-commutative operations are vital.

An Illustrative Example

The collection of invertible $2 \times 2$ matrices, whether with real or other types of coefficients like complex numbers, constitutes a non-abelian group under matrix multiplication.

An important clarification: only invertible matrices, i.e., $2 \times 2$ matrices with a nonzero determinant, are considered to form a group.

This specific set is referred to as the general linear group of degree 2, symbolized as $GL(2, \mathbb{R})$ when dealing with real coefficients.

It qualifies as a group by meeting these core properties:

Closure
The multiplication of any two invertible matrices yields another invertible matrix.
Associativity
The act of multiplying matrices is associative.
Identity Element
The identity matrix $I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ serves as the neutral element, ensuring $AI = IA = A$ for any matrix $A$ in $GL(2, \mathbb{R})$.
Inverse Element
Every invertible matrix $A$ possesses an inverse $A^{-1}$, enabling $AA^{-1} = A^{-1}A = I$.

The group $GL(2, \mathbb{R})$ showcases non-abelian characteristics, evident through the existence of at least two matrices $A$ and $B$ within $GL(2, \mathbb{R})$ such that $AB \neq BA$.

This principle illustrates how the order of multiplying matrices can alter the result, a notion that can be easily demonstrated. Take, for instance, matrices $A$ and $B$:

$$ A = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} $$

Upon calculating the products $AB$ and $BA$, we find:

$$ AB = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} $$

$$ BA = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix} $$

The difference in outcomes between $AB$ and $BA$ conclusively proves the non-abelian nature of the group of $2 \times 2$ invertible matrices with respect to multiplication.