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When a group is defined, we ask that the elements belong to a set. If we allow them to belong to a proper Class we get a GROUP.

What is the advantage of working with groups? What properties do we lose when we work with GROUPS?

An example of the notation can be found here:

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I would say that is a matter of historical development, in the sense that groups precede classes. And one disanvantage that comes to my head now, how do you define the quotient GROUP? (Since partitions of classes require to define something that it is not a class.) – Josué Tonelli-Cueto Nov 18 '11 at 17:11
People have found ample occupation when dealing with plain all small groups... What exactly do you want to achieve by considering GROUPS? What do you expect would be different in, say, your favorite textbook on group theory? – Mariano Suárez-Alvarez Nov 18 '11 at 17:47
@Mariano: Something pointed out to me today: let $V$ be the universe of all sets, then the permutation ‘group’ $\textrm{Sym}(V)$ in fact contains every (small) group as a subgroup. – Zhen Lin Nov 18 '11 at 19:18
In typical theories of (sets and) classes, all the PERMUTATIONS of $V$ won't even form a GROUP, because the individual PERMUTATIONS are proper classes and thus can't be members of a class. If we enlarge our world more, to allow collections of proper classes, then Sym(V) is such a super-class. It contains all class-sized groups (by Cayley) but not all super-class sized ones. – Andreas Blass Nov 11 '12 at 1:28
Whatever set- or class-theoretic level you allow for groups, you'll soon want higher levels, for (the natural construction of) quotient groups, automorphism groups, etc. So either work with sets, so that htese higher levels are available, or, if you really want proper classes, work in a theory that also allows super-classes, super-duper-classes, etc. I find it simplest to work in set theory. – Andreas Blass Nov 11 '12 at 1:31

For the same reasons why most of the time we only talk about sets and not proper classes in general. In vast majority of contexts, proper classes are not needed at all.

In practice, we're interested in small objects. Proper classes arise mostly when we want some kind of very universal object. But even then the universality is usually rather superficial – you can just think that the supposed proper class is actually a set in a proper extension of the universe, and its only universal for objects of small size. In that way, being a proper class is pretty relative.

A different problem is that for proper classes we don't have many constructions and theorems that are commonly used, like equivalence relations, quotients (so no isomorphism theorems!, no presentations!), wellorderings. If we want to add those things we're basically doing the same as what I mentioned in the previous paragraph: we make the universe larger.

I'm simplifying a bit, there are some subtleties involved, and many of these obstacles can be circumvented using some clever tricks, but generally proper classes are not important, nor do they happen too often if you aren't actually looking for them. If you want really big objects, you'd be better off assuming there's a sufficiently large strongly inaccessible cardinal and working in a restricted universe. That makes for far less of an ontological headache, and doesn't really make you lose anything of interest, as far as I can tell.

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