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This search for "folklore" in the tag suggests that this question is new to MSE.

I am aware that this might be too broad. If it is, I'm sorry. I have included the tag for good measure.

The Question:

What are some mathematical folklore theorems in the area of group theory?

Context:

What do I mean by "folklore"?

Well, according to Wikipedia,

In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists.

An example:

The only idempotent of a group $G$ is the identity element $e$.

Proof: Let $x^2=x\in G$. Then $xx=x^2=x=xe$, so, multiplying on the left by $x^{-1}$, we get $x=e$. $\square$

As far as I can tell, this theorem has no known originator. I think this is due to its simplicity. I don't recall how I came across it. According to @lhf, though, this is not folklore, so I'm not sure.

Why ask?

Because I think answers to this question will be valuable to the mathematical community at large, not just MSE; I am curious; and I don't want to miss out on things.

Please help :)

Shaun
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    Your example is weak. It’s a basic simple result. Not folklore – lhf Feb 05 '22 at 14:21
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    Fair enough, @lhf. I'm sorry. – Shaun Feb 05 '22 at 14:23
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    Please give a good example of what you have in mind for a folklore theorem – lhf Feb 05 '22 at 15:11
  • I don't know of any, @lhf. – Shaun Feb 05 '22 at 15:12
  • Is the fact that a subgroup of index $2$ is normal "folklore"? I've always said so, but I may have been misusing the term. – ancient mathematician Feb 05 '22 at 15:34
  • I'd have said, "yes" to that, @ancientmathematician, until lhf said my example wasn't folklore. – Shaun Feb 05 '22 at 15:38
  • Or "Groups of exponent $2$ are abelian." – ancient mathematician Feb 05 '22 at 15:49
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    How would you know that your examples are unpublished (as required by your definition)? They are basic and might appear as exercises or lemmas in some textbook or paper. (Not that I have any desire to search for these.) – Moishe Kohan Feb 05 '22 at 15:55
  • By being well-read, I suppose, @MoisheKohan; researchers face this problem too. – Shaun Feb 05 '22 at 15:57
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    Would this qualify? https://math.stackexchange.com/questions/40996/prove-that-if-abi-aibi-forall-a-b-in-g-for-three-consecutive-integers – Perhaps all interesting exercises in Herstein qualify :-) – lhf Feb 05 '22 at 16:21
  • I have never heard of a folklore theorem. An example of what I would consider to be mathematical folklore is the explanation of why there is no Nobel prize in mathematics. It is an unverified belief that this is because Nobel's wife had an affair with a mathematician. Another example would be the one about John Conway (or Hawking if you're a physicist) showing up late to a class and believing an unsolved problem was homework. – John Douma Feb 05 '22 at 16:24
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    Sorry for questioning your erudition and being well-read, but how many group theory books did you read in French, German and Russian? How many articles? Keep in mind that the subject goes back to 19th century. – Moishe Kohan Feb 05 '22 at 17:55
  • I wasn't referring to me, @MoisheKohan. Do you not think there are folklore theorems? – Shaun Feb 05 '22 at 17:58
  • @Shaun With your definition, the right answer is "impossible to know", as long as we are talking about basic group theory. If we are talking about modern, cutting edge group theory, then such examples are possible, short-term. I can give one example, going back to late 1990s. But it no longer qualifies as folklore. – Moishe Kohan Feb 05 '22 at 18:03
  • Does the classification of finite simple groups count?... – user1729 Feb 05 '22 at 18:20
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    @user1729 The authors of the classification of finite groups are well-known, and so are the references. Given Shawn's definition, this is not folklore. – Moishe Kohan Feb 05 '22 at 18:45
  • @JohnDouma The better questions (concerning nobel prizes) is why they are awarded at all ! Mathematics has similar prizes , for example the Fields Medal. The answer is probably that humans tend to make everything to a competation (musik, cooking , dancing , etc.). The merit of such competitions is debatable – Peter Feb 23 '22 at 11:16

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