Linked Questions

72
votes
5answers
2k views

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,...
11
votes
6answers
9k views

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 subgroup of the group of permutations on countably many symbols, hence the latter ...
1
vote
1answer
2k views

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 ${\cal P}(X)$?