Group with elements other than numbers

a) Permutation Group:

A permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G. A Permutation group element can be rearranged as shown as below.

G = {H, T} HT or TH

b) Matrix Group:
A matrix group is a group G consisting of invertible matrices over a specified field K, with the operation of matrix multiplication. A linear group is an abstract group that is isomorphic to a matrix group over a field K.

For more information:

a) Permutation group:

Suppose we toss a coin. We then get either head H or a tail T. Consider the set G={H,T} and the ordered rearrangements or permutations of its elements .Obviously , only two cases arise:  H T or T H .

In other words , we  may ' replace head H by head H and tail T by tail T in the first case'; and 'replace head H by tail T and tail T by head H' in the second case .

            This situation is represented symbolically in the following way:

and 

Consider the set S_G  ={σ,µ} and define the binary operation * on S_G as  the product or composition ' σµ' or 'σ *µ' of the two permutation as ' One permutation σ followed by another permutation  µ ' or 'Do σ and then µ'.

which shows σ is both an identity element and inverse of itself.

which confirms σ   is an identity element.

which shows µ is the inverse of itself.

b) Matrix group:  

The set of square matrices of a given order is known to possess the associative property , the zero matrix as the identity matrix and the negative of a matrix as the inverse of the matrix. This means the set of all square matrices of a given  order is a group .Such a group is known as the matrix group of the given order .A matrix group of order 2 under matrix addition is generally denoted by:

        Furthermore, the set of non-singular square matrices of a given  order under matrix multiplication is known to be associative , has the unit matrix as the identity matrix , and the inverse of the given  matrix as the inverse element .this means the set of non-singular square matrices forms a group under matrix multiplication.