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.