Question

Let T be a nonempty set, and A(T) the set of all perμtations of T. Show that A(T) is a group under the operation of composition of functions.

a) Closure property
b) Associative property
c) Identity element
d) Inverse element

Answer 1

To show that A(T) is a group under the operation of the composition of functions, we need to prove the closure, associative, identity, and inverse properties.

Step-by-step explanation:

To show that A(T) is a group under the operation of composition of functions, we need to prove the following properties:

1. Closure property: For any two permutations f and g in A(T), their composition f ∘ g is also a permutation of T. This can be shown by observing that f and g are bijective functions and their composition will also be bijective, satisfying the conditions of a permutation.
2. Associative property: For any three permutations f, g, and h in A(T), the composition is associative, meaning (f ∘ g) ∘ h = f ∘ (g ∘ h). This can be proved by considering the composition of functions and their associativity.
3. Identity element: There exists an identity permutation, denoted by e, such that for any permutation f in A(T), e ∘ f = f ∘ e = f. The identity permutation maps each element of T to itself.
4. Inverse element: For every permutation f in A(T), there exists an inverse permutation denoted as f^(-1), such that f ∘ f^(-1) = f^(-1) ∘ f = e. The inverse permutation undoes the mapping of f.