Let a mapping f : G→ G defined by f (x) = x-1 , then which of the following is / are true ?a)f is one - oneb)f is on toc)f is homomorphism as well as automorphismd)G is a non - abelian group.Correct answer is option 'A,B,C'. Can you explain this answer?

Question

Let a mapping f : G&rarr; G defined by f (x) = x-1 , then which of the following is / are true ?a)f is one - oneb)f is on toc)f is homomorphism as well as automorphismd)G is a non - abelian group.Correct answer is option 'A,B,C'. Can you explain this answer?

Answer

F : G→ G defined by

Let G be a commutative group and

Hence F : G→G is a isomorphism.

So f is an automorphism.

Answer

Maps a group G to a group G'. Then, for all elements a, b in G, the mapping f satisfies the following properties:

1. f(a * b) = f(a) * f(b) (Closure under group operation)
2. f(e) = e' (Preservation of identity element)
3. f(a^-1) = (f(a))^-1 (Preservation of inverses)

These properties ensure that the mapping f preserves the group structure of G.

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