TITLE: Why is the collection of all groups a proper class rather than a set?
QUESTION [25 upvotes]: According to Wikipedia,

The collection of all algebraic objects of a given type will usually
be a proper class. Examples include the class of all groups, the class
of all vector spaces, and many others. In category theory, a category
whose collection of objects forms a proper class (or whose collection
of morphisms forms a proper class) is called a large category.

I am aware of Russell's Paradox, which explains why not everything is a set, but how can we show the collection of all groups is a proper class?

REPLY [35 votes]: The collection of singletons is not a set. Therefore the collection of all trivial groups is not a set.
If you wish to consider "up to isomorphism", note that for every infinite cardinal $\kappa$ you can consider the free group, or free abelian group with $\kappa$ generators. These are distinct (up to isomorphism, that is), and since the collection of cardinals is not a set the collection of groups cannot be a set either.