I Ways of writing a logical argument

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Let ##G## be a group. Suppose that the map from ##G## to itself defined by ##\phi (g) = g^{-1}## is a homomorphism. Prove that ##G## is abelian.

So I came up with two ways of writing the solution and am wondering whether they are equivalent and which one is preferable:
1) Let ##a,b \in G##. Then ##\phi (ab) = \phi(a)(b) \implies (ab)^{-1} = a^{-1}b^{-1} \implies b^{-1}a^{-1} = a^{-1}b^{-1} \implies ba = ab##.

2) Let ##a,b \in G##. Then ##ab = (b^{-1}a^{-1})^{-1} = \phi(b^{-1}a^{-1}) = \phi(b^{-1}) \phi(a^{-1}) = b a##
 

fresh_42

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It doesn't matter, they are both valid. I found the first one easier to read (you've forgotten one ##\phi##), but this is a matter of taste.

The basic idea is contained in both of them: inversion reverses the order whereas a homomorphism preserves the order. Both at the same time forces commutativity.
 

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