Is my proof correct? (Abstract Algebra - Groups)

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In summary, the conversation discusses a proof for a problem in abstract algebra involving finite groups of even order. The attempt at a solution uses a relabeling technique to show that there must be an element in the group that is equal to its inverse, proving the given statement. The speaker asks for someone else to check their proof and suggests that it may be an unusual approach.
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Is my proof correct? (abstract algebra - groups)

Homework Statement


If G is a finite group of even order, show that there must be an element a≠e such that a=a-1
I believe my proof is a bit odd and unusual, I'd appreciate it if someone else checks it and suggests a more convenient argument for this problem.

The Attempt at a Solution


well, since G is a finite group of even order, let's assume |G|=2k. since G is finite, we can assume G looks like this: [itex]G=\{e,a,a^{-1},b,b^{-1},ab,(ab)^{-1},...\}[/itex]
But if we relabel all elements, we can show G in the form: [itex]G=\{e,g_1,g_1^{-1},...,g_k,g_k^{-1}\}[/itex], let's call this new representation of G as G' and notice that G=G'. if we exclude e, we have |G-{e}|=2k-1. the number of [itex]g_i[/itex]'s in G' is k, so if all their respective [itex]g_i^{-1}[/itex]'s were distinct, G'-{e} would have 2k elements, but that would be impossible because G and G' were the same set! so that would mean that not all [itex]g_i[/itex]'s and [itex]g_i^{-1}[/itex] are distinct, so there exists a [itex]g_i[/itex] for which we have: [itex]g_i[/itex]=[itex]g_i^{-1}[/itex] Q.E.D
 
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Seems good to me!

I think the argument is a really nice one. Maybe it can be rephrased a little bit here and there. But I think that it's ok.
 

1. What is the first step in verifying the correctness of a proof in Abstract Algebra?

The first step in verifying the correctness of a proof in Abstract Algebra is to carefully read through the proof and make sure that each step follows logically from the previous one.

2. How can I check if my proof is using the correct definitions and theorems?

To check if your proof is using the correct definitions and theorems, you can consult a textbook or other reliable sources to confirm that your reasoning is in line with accepted mathematical principles.

3. Is it important to include all necessary steps in my proof?

Yes, it is important to include all necessary steps in your proof to ensure that your argument is complete and logically sound. Leaving out steps or assuming certain facts without justification can lead to an incorrect proof.

4. Should I use specific examples to illustrate my proof?

It can be helpful to use specific examples to illustrate your proof, but it is important to remember that your proof should hold true for all cases, not just the specific examples you use.

5. How can I make sure my proof is well-written and easy to understand?

To make sure your proof is well-written and easy to understand, you can ask a colleague or mentor to review it and provide feedback. It can also be helpful to read your proof aloud to yourself to check for any confusing or unclear parts.

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