Class of groups

A class of groups is a set-theoretical collection of groups satisfying the property that if G is in the collection then every group isomorphic to G is also in the collection.

This concept arose from the necessity to work with a bunch of groups satisfying certain special property (for example finiteness or commutativity).

Since set theory does not admit the "set of all groups", it is necessary to work with the more general concept of class.

A class of groups

is a collection of groups such that if

Groups in the class

For a set of groups

the smallest class of groups containing

denotes its isomorphism class.

The most common examples of classes of groups are: Given two classes of groups

it is defined the product of classes This construction allows us to recursively define the power of a class by setting It must be remarked that this binary operation on the class of classes of groups is neither associative nor commutative.

For instance, consider the alternating group of degree 4 (and order 12); this group belongs to the class

has no non-trivial normal cyclic subgroup, so

However it is straightforward from the definition that for any three classes of groups

, A class map c is a map which assigns a class of groups

to another class of groups

A class map is said to be a closure operation if it satisfies the next properties: Some of the most common examples of closure operations are: