Ways of writing a logical argument

So they are equivalent.In summary, the map from ##G## to itself defined by ##\phi (g) = g^{-1}## being a homomorphism implies that ##G## is abelian, as shown by the fact that for any ##a,b \in G##, ##ab = b a##.
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Mr Davis 97
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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##
 
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  • #2
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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1. What are the key elements of a logical argument?

The key elements of a logical argument include a clear and debatable claim, supporting evidence, logical reasoning, counterarguments, and a conclusion.

2. How do I structure a logical argument?

A logical argument is typically structured as follows: introduction (including the claim), body paragraphs (each with supporting evidence and reasoning), counterarguments, and conclusion (restating the claim and summarizing the main points).

3. What are some common flaws to avoid in a logical argument?

Some common flaws to avoid in a logical argument include using emotional appeals instead of logical reasoning, relying on biased or insufficient evidence, making generalizations or assumptions, and ignoring counterarguments.

4. How can I make my logical argument more persuasive?

To make your logical argument more persuasive, you can use a variety of techniques such as using strong and relevant evidence, anticipating and addressing counterarguments, using logical fallacies to strengthen your argument, and using persuasive language and tone.

5. Is there a specific format or style to follow when writing a logical argument?

While there is no specific format or style to follow, it is important to use a clear and organized structure, provide evidence to support your claims, use logical reasoning, and engage with potential counterarguments. Additionally, following a proper citation style and using clear and concise language can also enhance the effectiveness of your argument.

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