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Thread: Proposition Regarding Cosets and Orders

  1. #1
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    Proposition Regarding Cosets and Orders

    Let $\displaystyle A$ be a finite subset of a group $\displaystyle G$ (not necessarily a subgroup). Denote by $\displaystyle A^2$ the set $\displaystyle \{a_1 a_2 | a_1, a_2 \in A\}$.

    Prove that $\displaystyle |A^2| = |A| \iff $ the following is true:
    $\displaystyle A$ equals a left coset $\displaystyle aH$ for some subgroup $\displaystyle H \leq G$ and some
    element $\displaystyle a \in G$, and $\displaystyle A$ also equals some right coset $\displaystyle Hb$, $\displaystyle b \in G$.

    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!
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  2. #2
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    I also realize that the professor went through quite a bit about group actions, would they be required to give a proof of this?
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  3. #3
    Senior Member roninpro's Avatar
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    I haven't worked this out completely, but I'll throw out some ideas.

    Let $\displaystyle A=\{a_1, a_2,\ldots, a_n\}$ and write $\displaystyle A^2=\bigcup_{a\in A} aA$. Since $\displaystyle a\in G$, the operation $\displaystyle x\mapsto ax$ is injective. Therefore, $\displaystyle |aA|=|A|$ for every $\displaystyle a\in A$. Now, we are given that $\displaystyle |A^2|=|A|$. This implies that every term in the union is equal; that is, $\displaystyle a_1A=a_2A=\ldots=a_nA$.

    This seems to be coset-like behaviour, but I don't quite see to how get the conclusion.
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  4. #4
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    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 $\displaystyle a^{-1}A$ for any $\displaystyle a \in A$. Then $\displaystyle A \subset a^{-1}A^{2}$. From there, a few further arguments will show that $\displaystyle a^{-1}A$ is a subgroup, and it is clear that $\displaystyle A = a.a^{-1}A$.
    Last edited by h2osprey; Nov 6th 2010 at 03:36 PM.
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