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
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.