Theorem:-7show that a group G is an abelian iff f(a)= a inverse is an automorphism.(for8marks#bbmku)

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f(a)=inverse of a is an automorphism iff G is abelian. f(a)=inverse of a is an automorphism iff G is abelian.
Let G be a group and Phi an automorphism of G. If a in G has order o(a), then o(phi(a)) is o(a) . Let G be a group and Phi an automorphism of G. If a in G has order o(a), then o(phi(a)) is o(a) .
Proof that f(a) = a^(-1) is a Group Isomorphism if G is Abelian Proof that f(a) = a^(-1) is a Group Isomorphism if G is Abelian
Let G be an abelian group. Then Φ:G→G given by .... is an automorphism. Let G be an abelian group. Then Φ:G→G given by .... is an automorphism.
a↦a^(-1) is an automorphism of a group G iff G is abelian a↦a^(-1) is an automorphism of a group G iff G is abelian
Proof: Group Element is the Inverse of its Inverse | Abstract Algebra Proof: Group Element is the Inverse of its Inverse | Abstract Algebra
Show that a group 𝐺 is an abelian iff (ab)^2=a^2b^2 ,for all a,b∈𝐺 Show that a group 𝐺 is an abelian iff (ab)^2=a^2b^2 ,for all a,b∈𝐺
Proove that a group G is Abelian iff (ab) ^ - 1 =a^ -1 b^ -1 for all a,b in G Proove that a group G is Abelian iff (ab) ^ - 1 =a^ -1 b^ -1 for all a,b in G