I also realize that the professor went through quite a bit about group actions, would they be required to give a proof of this?
Let be a finite subset of a group (not necessarily a subgroup). Denote by the set .
Prove that the following is true:
equals a left coset for some subgroup and some
element , and also equals some right coset , .
The backward implication is easy enough since all that's required is a simple bijection, but I'm having problems proving the forward implication.
Thanks in advance!
I haven't worked this out completely, but I'll throw out some ideas.
Let and write . Since , the operation is injective. Therefore, for every . Now, we are given that . This implies that every term in the union is equal; that is, .
This seems to be coset-like behaviour, but I don't quite see to how get the conclusion.
Solved it -- took much more effort than I thought was initially required, presumably because this question was adapted from a conference paper that's currently investigating groups with such properties.
Basically the proof shows that the identity element is in for any . Then . From there, a few further arguments will show that is a subgroup, and it is clear that .