Are infinite groups permutation groups? I have known Cayley's Theorem for some time now, which shows that all finite groups are permutation groups (secretly, as a previous mathematics teacher of mine might have put it). However, the thought occurs to me whether we can prove that this is true for infinite groups. 
In this amazing MO answer, they show what appears to a construction of what I now call the Mother Class-Group, but only implicitly suggest that this contains all other groups as sub-groups. Why is it? Is it to do with the fact that all finite groups are linear?
 A: Just glancing at that MO post, here's an analogous construction of a mother of all groups that's perhaps simpler, and is in any case quite explicit. Note All I'm claiming here is that it's a "proper-class group" that contains every group as a subgroup; in particular I'm not claiming that it does the other mode-theoretic things that the "monster group" is said to do in that MO post. (Edit: Turns out it was a little too simple, running into  some illegal set theory. See Oops below...)
Let  $O$ be the class of all ordinals. Let $M$ be the class of all bijections $\phi:O\to O$ such that there exists $\alpha$ with $\phi(\beta)=\beta$ for all $\beta>\alpha$.
Then $M$ is what they call a "proper-class group", which is to say it's "a group except that it's a proper class, too large to be a set".
It's easy to see that if $G$ is a genuine "set-sized" group then $G$ is isomorphic to a subgroup of $M$: The proof of Caley's theorem shows that $G$ is isomorphic to a subgroup of the bijections of some set $S$, and there exists an ordinal $\alpha$ such that there exists a bijection of $S$ onto $\alpha$; the group of bijections of $\alpha$ is a subgroup of $M$.
Oops. The elements of $M$ above are themselves proper classes; that's not allowed, proper classes are not supposed to be elements.
Almost fix: If $\phi:\alpha\to\alpha$ is a bijection then $\phi$ defines a bijection $\psi\to O$ such that $\psi(\beta)=\beta$ for all $\beta\ge\alpha$. So redefine $M$ to be the class of all bijections from some ordinal to itself; define the composition of two such things to be the $\phi$ that corresponds to the composition of the corresponding $\psi$s. (That is, $\psi_{\phi_1\phi_2}=\psi_{\phi_1}\circ\psi_{\phi_2}$.)
That doesn't quite work either, because one $\psi$ arises from many different $\phi$s. Obvious solution is to define an equivalence relation blah blah - oops, the equivalence classes are again proper classes. But for each $\psi$ there is a minimal $\phi$.
fix: Say $S_\alpha$ is the set of all bijections $\phi:\alpha\to\alpha$ such that the set of $\beta\in\alpha$ with $\phi(\beta)\ne\beta$ is cofinal. Now any nontrivial bijection $\psi:O\to O$ arises from a unique $\phi\in\bigcup_\alpha S_\alpha$. Let $M=\{\emptyset\}\cup\bigcup_\alpha S_\alpha$. (Now $\emptyset$ becomes the identity element of $M$.)
