This search for "folklore" in the group-theory 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 big-list 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In common mathematical parlance, a mathematical result is called folklore if it is an unpublished result with no clear originator, but which is well-circulated and believed to be true among the specialists.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 :)