Let G be a group and f: G→ G be a mapping defined by () = −1 ∀ x ∈ G. Then f is an automorphism if and only if i) G is commutative ii) G is non-commutative iii) G is a finite cyclic group iv) G≠ {e}, e is the identity element of G​

The statement is asking to determine under what conditions the function f: G→ G defined by f(x) = x^-1 for all x in G is an automorphism. An automorphism is an isomorphism from a mathematical object to itself. In the context of groups, it is a bijective function that preserves the group operation.

i) If G is commutative (i.e., for all a, b in G, ab = ba), then f is an automorphism. This is because for all a, b in G, f(ab) = (ab)^-1 = b^-1 * a^-1 = f(b) * f(a), which shows that f preserves the group operation.

ii) If G is non-commutative (i.e., there exist a, b in G such that ab ≠ ba), then f is not necessarily an automorphism. This is because for some a, b in G, f(ab) = (ab)^-1 = b^-1 * a^-1, which is not necessarily equal to f(b) * f(a) = a^-1 * b^-1.

iii) If G is a finite cyclic group, then G is commutative, so f is an automorphism by the argument in (i).

iv) If G ≠ {e}, where e is the identity element of G, then f is not necessarily an automorphism. This is because the condition G ≠ {e} does not provide enough information to determine whether f preserves the group operation. For example, if G is a non-commutative group, then f is not an automorphism by the argument in (ii).