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Just ANSWER THE QUESTION

Here's a Math SE pathology that bugs me. OP will ask "I'm trying to prove that groups !!A!! and !!B!! are isomorphic, I constructed this bijection but I see that it's not a homomorphism. Is it sufficient, or do I need to find a bijective homomorphism?"

And respondent !!R!! will reply in the comments "How can a function which is not an homomorphism prove that the groups are isomorphic?"

Which is literally the exact question that OP was asking! "Do I need to find … a homomorphism?"

My preferred reply would be something like "Your function is not enough. You are correct that it needs to be a homomorphism."

Because what problem did OP really have? Clearly, their problem is that they are not sure what it means for two groups to be isomorphic. For the respondent to ask "How can a function which is not an homomorphism prove the the groups are isomorphic" is unhelpful because they know that OP doesn't know the answer to that question.

OP knows too, that's exactly what their question was! They're trying to find out the answer to that exact question! OP correctly identified the gap in their own understanding. Then they formulated a clear, direct question that would address the gap.

THEY ARE ASKING THE EXACT RIGHT QUESTION AND !!R!! DID NOT ANSWER IT

My advice to people answering questions on MSE:

It's all very well for !!R!! to imagine that they are going to be brilliant like Socrates, conducting a dialogue for that ages that draws from OP the realization that the knowledge they sought was within them all along. Except:

1. !!R!! is not Socrates
2. Nobody has time for this nonsense
3. The knowledge was not within them all along

MSE is a site where people go to get answers to their questions. That is its sole and stated purpose. If !!R!! is not going to answer questions, what are they even doing there? In my opinion, just wasting everyone's time.

Important pedagogical note

It's sufficient to say "Your function is not enough", which answers the question.

But it is much better to say "Your function is not enough. You are correct that it needs to be a homomorphism". That acknowledges the student's contribution. It tells them that their analysis of the difficulty was correct!

They may not know what it means for two groups to be isomorphic, but they do know one something almost as good: that they are unsure what it means for two groups to be isomorphic. This is valuable knowledge.

This wise student recognises that they don't know. Socrates said that he was the wisest of all men, because he at least “knew that he didn't know”. If you want to take a lesson from Socrates, take that one, not his stupid theory that all knowledge is already within us.

OP did what students are supposed to do: they reflected on their knowledge, they realized it was inadequate, and they set about rectifying it. This deserves positive reinforcement.

Addenda

1. This is a real example. I have not altered it, because I am afraid that if I did you would think I was exaggerating.

2. I have been banging this drum for decades, but I will cut the scroll here. Expect a followup article.

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