On the meaning of "Class of finite groups".

Question

What do we mean precisely when we speak about a class of finite groups? Is this simply a collection of some finite groups, maybe collected with a criterion (example: the class of all finite cyclic groups), or there exists a precise definition of "class of finite groups" involving some other properties?

In the case a class of finite groups is simply a collection of finite groups which follows some criterion (ex finite cyclic groups as above), what can we say about a collection of some finite groups collected randomly? Ex $\{S_5, A_3,D_4, C_9\}$... I'd call it simply a set of finite groups. Is this right?

Thank you all

Answer 1

IF the collection $\mathcal G$ of all finite groups $(G,\times)$ is a set, then you can define a set:

$$\mathcal S = \bigcup_{(G,\times)\in \mathcal G} G$$

In particular, if $Y$ is any set, there is a group on the singleton set $G=\{Y\}$. So for any set $Y$, $Y\in \mathcal S$. That means $\mathcal S$ contains all sets. That is not allowed in set theory because of the standard paradoxes.

If you take a given infinite set, $X$, you can define all groups on finite subsets of $X$, and then that collection is a set. And every finite group is isomorphic to one of these groups on subsets of $X$, but the isomorphism is neither unique or "natural," so it can be tricky, in category theory, to deal with that fact.

There is also a confusion in the usage of the term "class" here, since, when dealing with finite groups, we talk about the classification, and that means something different than the term "class" from set theory.

Answer 2

The standard definition of a class of groups is a a class $\mathcal{C}$ (in the set-theoretic sense) whose members are groups with the property that if $G$ and $H$ are isomorphic groups then $G \in \mathcal{C} \Leftrightarrow H \in \mathcal{C}$.

So yes, there is a class of finite cyclic groups and a class of nilpotent groups, etc. It is customary to study the closure properties of specific classes. For example, a class may or may not be closed under subgroups, quotient groups, direct products, free products, etc.