# Group of Permutations

The set of all permutations on symbols is a finite group of order with respect to the composite of mappings as the operation. For , this group is abelian and for it is always non-abelian.

Let be a finite set having distinct elements. Thus there are permutations possible on . If denotes the set of all permutations of degree then the multiplication of permutation on satisfies the following axioms.

__Closure Axiom__**:** Let , then each of them is one-one mapping of onto itself and therefore their composite mapping is a one-one mapping of onto itself. Thus is a permutation of degree on , i.e.

This shows that is closed under multiplication.

__Associative Axiom__**: **Since the product of two permutations on a set is nothing but the product of two one-one onto mappings on and the product of mapping is associative, the product of permutations also obeys the associative law. Hence

__Identity Axiom__**:** Identity permutation is the identity of multiplication in because

__Inverse Axiom__**:** Let then is one-one mapping, hence it is investible. Hence , the inverse mapping of is also one-one and onto. Consequently, is also a permutation in .

Thus the symmetric set of all permutations of degree defined on a finite set forms a finite group of order `\(n\)`

with respect to the composite of permutations as the composition.

__Commutative Axiom__**:** If we consider the symmetric group of permutations of degree 1 with respect to permutation product 0, then it consists of a single permutation, namely the identity permutation . Since , is an abelian group. If we consider the symmetric group of all permutations of degree 2, i.e. the group of all permutations defined on a set of two elements , then

Now

and

Therefore an operation having commutative is an abelian group of order 2. But when then the permutation product is not necessarily commutative. Hence is not necessarily an abelian group.