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$X$ is a set and $X!$ is the set of bijections from $X$ to $X$. Let $A(m)$ be the set defined as $A(m)=\{f^x(m)\mid x\in\mathbb Z\}$ for some $f$ in $X!$ and some $m$ in $X$. Now, how do I prove that for $m, n\in X$, $A(m)\cap A(n)=\emptyset$ or $A(m) = A(n)$?

I have tried the following strategy. Assume $A(m) \cap A(n) \ne \emptyset$. Then, there exists $x \in A(m) \cap A(n)$, which implies $x \in A(m)$ and $x \in A(n)$. Then, I think we have to use the set membership criteria. What I am not sure is how does that help us use the double containment argument to prove that $A(m) = A(n)$? To me, it seems very trivially right. I am not sure how to proceed in these kinds of questions.

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  • $\begingroup$ You perhaps need that $X$ is finite? $\endgroup$ – enedil Feb 10 at 20:54
  • $\begingroup$ I do not think so. $\endgroup$ – Ufomammut Feb 10 at 20:55
  • $\begingroup$ Ah, $x$ in the definition of $A$ can be negative, right? $\endgroup$ – enedil Feb 10 at 20:56
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    $\begingroup$ No, that is false. Consider $X=\mathbb Z$ and $f(x)=x+1$. This is a bijection, but doesn't have this property. $\endgroup$ – enedil Feb 10 at 20:58
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    $\begingroup$ Just a word on notation. Using $X$ for a set, and then $x$ for an element of $\Bbb Z$ is awful. Even worse, $m$ is a common letter for denoting an element of $\Bbb Z$, and you use it to denote an element of $X$ (I suppose?) $\endgroup$ – Asaf Karagila Feb 10 at 21:10
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I don't think I'm understanding the definition as written. It appears that you've written $A(m)=\{f^x(m): x\in\Bbb{Z}, f\in X!\}$. Do you perhaps mean to instead say: Fix $f\in X!$, then for each $m\in X$, we define $A(m)=\{f^x(m):x\in\Bbb{Z}\}$? It must be the latter I think.

In that case, this is a special case of the fact that orbits under a group action are either equal or disjoint, which in this case is proved in the following manner.

Suppose $A(m)\cap A(n)\ne \varnothing$. Let $f^x(m)=f^y(n)$. Then $m=f^{y-x}(n)$, so $f^k(m)=f^{y-x+k}(n)\in A(n)$. Thus $A(m)\subseteq A(n)$, and by symmetry, we also have $A(n)\subseteq A(m)$. Thus $A(m)=A(n)$.

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  • $\begingroup$ Yes, the $f$ is fixed. What I do not understand is in your solution, why did you say "Let $f^x(m) = f^y(n)$." I mean I got the same, but in following way: If $A(m)$ $\cap$ $A(n)$ $\neq$ $\emptyset$, then there exists an $a$ $\in$ $X$ such that a $\in$ $A(m)$ $\cap$ $A(n)$, and then I concluded that $f^x(m) = f^y(n)$. Is that your way as well? $\endgroup$ – Ufomammut Feb 10 at 21:08
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    $\begingroup$ @Ufomamut, yes, that's the same. Since we know that $A(m)\cap A(n)$ is nonempty, then there is an element in $A(m)$ that equals an element of $A(n)$, and those elements have the form $f^x(m)$ and $f^y(n)$ for some $x$ and $y$ in $\Bbb{Z}$. $\endgroup$ – jgon Feb 10 at 21:11

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