The set of all permutations on symbols is a finite group of order with respect to 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 multiplication of permutation on satisfy 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 being associative, the product of permutations also obeys the associative law. Hence
Identity Axiom: Identity permutation is 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 permutation of degree defined on a finite set forms a finite group of order
\(n\) with respect to the composite of permutation 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 permutation of degree 2, i.e. the group of all permutations defined on a set of two elements , then
Therefore operation having commutative is Abelian group of order 2. But when then permutation product is not necessarily commutative. Hence then is not necessarily an Abelian group.