$\newcommand{\Sym}{\operatorname{Sym}}$ I was studying $\Sym(\mathbb{N})$, the set consisting of all the bijections from $\mathbb{N}$ to itself. Since it is a group, the concept of "period of an element" has a sense, and it is the smallest positive integer $n$ such that $f^n = e$, where $f$ is one of those bijections and $e$ is the identity of the group (the identity function).

I was interested in the subset of all the elements of the group that have finite period. My question is: if I randomly choose an element of $\Sym(\mathbb{N})$, is there a way to know... if it's more likely to get an element of infinite period, or an element of finite period? The problem is that, according to the results I got, both $\Sym(\mathbb{N})$ and its subset I'm interested in are infinite sets that have the cardinality of the continuum.

Am I unawarely asking a stupid/impossible question, or are there any mathematical tools to know what that probability is?

  • $\begingroup$ Very nice question. Are you familiar with measure theory? It may be helpful in answering your question here. For example, measure theory can be used to answer the (similar sounding) question "What is the probability of randomly selecting a rational number from the real numbers?" As you might expect (because $\mathbb{Q}$ is countable), the answer is $0$, an intuition which can be formalized by noting that the rationals are a measure $0$ subset of the reals. I'm far from knowledgeable in this area though, so I must confess I don't know if it will be useful but my guess would be to start there. $\endgroup$ – Alex Wertheim Aug 23 '14 at 23:04
  • $\begingroup$ @AWertheim I know some things about it, but only a few basic concepts; if measure theory is a tool that can be used to find an answer here, I understand why my attempts at finding a solution were all unsuccessful... I did not think of it because I did not (and still do not) know it well enough to use it to solve such problems. $\endgroup$ – Labba Aug 23 '14 at 23:11

Maybe the hardest part of answering this question is deciding whether there's some natural meaning of the concept of randomly choosing an element of this set.

The set in question is uncountably infinite.

Under the most usual conventions of probability, one wants a probability measure on some sigma-algebra of subsets of the space in question. Which sigma-algebra should be used in this case? Is there some natural answer to that question. Should we regard the set of all permutations that map $3$ to $8$ (and others like that) as a sub-basic open set, and the set of all unions of finite intersections of such sets as open, and then look at Borel sets in that topological space?

After that, which probability measure on the sigma-algebra should we assign?

What, for example, is the probability that $3$ is mapped to $8$? If $\tau$ is a randomly chosen permutation, then certainly $$ \sum_{n\in\mathbb N} \Pr(\tau(3)=n) = 1. $$ But the probabilities $\Pr(\tau(3)=n)$ for $n\in\mathbb N$ would differ from each other.

Maybe there are lots of interesting probability distributions on this set, just as on the sigma-algebra of Borel subsets of $\mathbb R$. We all learn about the normal distribution, the exponential distribution, etc., etc., etc.

So there's still a lot of work to do before you have a well-defined mathematical question.

  • $\begingroup$ I see; anyway, thank you very much for your answer here. $\endgroup$ – Labba Aug 24 '14 at 0:31

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