Set of All Groups

logic set-theory group-theory

In my undergraduate Group Theory class, while discussing the impossibility of equivalence relations on groups, my professor said that the set of all groups does not exist.

Are there any group subcategories for which the set of all such groups DO exist? As examples,

i) Does the set of all cyclic groups exist?

ii) Does the set of all finite groups exist?

iii) Does the set of all groups of order n, for some integer n, exist?

Thank you.


Yes and no. The underlying set of a trivial group can be any one-element set and the class of all one-element sets is a proper class (i.e. not a set). However, if we consider only groups up to isomorphism, then the set of cyclic groups exists (there is one for each natural numbre and there is $\mathbb Z$). In the same sense the set of groups of order $n$ (or more easily: The set of operations on a given set of $n$ elements such that this makes it a group) exists, and also the set of all finite groups.

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