# Linked Questions

4 questions linked to/from Cardinality of the permutations of an infinite set

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### When the automorphism group of an object determines the object

Let me start with three examples to illustrate my question (probably vague; I apologize in advance).
$\mathbf{Man}$, the category of closed (compact without boundary) topological manifold. For any $M,...

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### Easy proof of the uncountability of bijections on natural numbers

Is there an easy proof of the uncountability of bijections on natural numbers?
The proof that I have in mind is as follows -
$\text{Gal }(\overline{\mathbb Q}/\mathbb Q)$ is a proper uncountable ...

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### Terminology for a bijection from a set to itself

A current project uses bijections from a set to itself. (The set is the integer compositions of $n$, i.e., "ordered partitions of $n$," but that doesn't seem pertinent to the question.) Is ...

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### Cardinality of symmetric group [duplicate]

Possible Duplicate:
Cardinality of the permutations of an infinite set
Why does the symmetric group on an infinite set X have the cardinality of the power set ${\cal P}(X)$?