Non-Abelian Groups
What Is a Non-Abelian Group?
A non-Abelian group is a group (S,*) where the operation * does not satisfy the commutative property.
It's also known as a non-commutative group.
The term is named after the Norwegian mathematician Niels Henrik Abel.
A Practical Example
The set of invertible 2x2 real matrices, denoted by M2, forms a group (M2, ·) under the operation of matrix multiplication ( · ).
$$ (\ M_2\ , \ \cdot\ )\ \ \ \ where \ \ M_2 = \text{invertible 2x2 matrices} $$
This is called a multiplicative group because:
- The product of two 2x2 matrices is another 2x2 matrix. $$ \forall \ A,B \in M_2 \rightarrow AB \in M_2 $$
- The product of matrices satisfies the associative property. $$ A \cdot (B \cdot C) = (A \cdot B) \cdot C $$
- There is an identity element for multiplication, which is the identity matrix (I). $$ A \cdot I = A \cdot \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} = A $$
- Every invertible matrix A∈M2 has an inverse, denoted by A-1. $$ A \cdot A^{-1} = I $$
Note: In this example, we’re focusing solely on invertible matrices, which are matrices with a non-zero determinant. This condition is crucial because it ensures that every matrix has an inverse. If we included 2x2 matrices with a zero determinant (non-invertible matrices), some matrices in M2 would not have an inverse, violating one of the essential properties of groups.
The group (M2, ·) is non-Abelian because matrix multiplication is not commutative.
For instance, consider these two 2x2 matrices:
$$ A = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} $$
$$ B = \begin{pmatrix} -1 & 3 \\ 2 & 1 \end{pmatrix} $$
The product of these matrices, AB, is:
$$ AB = \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} \cdot \begin{pmatrix} -1 & 3 \\ 2 & 1 \end{pmatrix} = \begin{pmatrix} 3 & 5 \\ 5 & 6 \end{pmatrix} $$
If we reverse the order of multiplication, the product BA gives us a different result:
$$ BA = \begin{pmatrix} -1 & 3 \\ 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 1 & 2 \\ 1 & 3 \end{pmatrix} = \begin{pmatrix} 2 & 7 \\ 3 & 7 \end{pmatrix} $$
Since the results differ, the group (M2, ·) is a non-Abelian group.
$$ AB \ne BA $$
It only takes one instance of a differing result to demonstrate that, in general, matrix multiplication is not a commutative operation.
Note: On the other hand, the group of matrices (M2,+) under addition (+) is an Abelian group because addition is commutative. $$ \forall \ A,B \in M_2 \rightarrow A+B = B+A $$
And so on.
