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This is not a homework problem. I was just wondering.
Let G be a group and let A be a finite subset of G. If |A²|=|A|² (where [tex]A^2=\{a_1a_2~\vert~a_1,a_2\in A\} [/tex] ). Is it true that A is a left coset of G?
If A has two elements, then I have proven that this is true. But for greater elements, it soon becomes very complicated. I do think this is true...
Anybody got a proof/counterexample or maybe some hints?
Let G be a group and let A be a finite subset of G. If |A²|=|A|² (where [tex]A^2=\{a_1a_2~\vert~a_1,a_2\in A\} [/tex] ). Is it true that A is a left coset of G?
If A has two elements, then I have proven that this is true. But for greater elements, it soon becomes very complicated. I do think this is true...
Anybody got a proof/counterexample or maybe some hints?