# Why do we use groups and not GROUPS?

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: http://www.mathe2.uni-bayreuth.de/stoll/papers/games12.pdf

• 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.) Commented Nov 18, 2011 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? Commented Nov 18, 2011 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. Commented Nov 18, 2011 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. Commented Nov 11, 2012 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. Commented Nov 11, 2012 at 1:31