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

Definition

A class of groups is a collection of groups such that if and then. Groups in the class are referred to as -groups.
For a set of groups, we denote by the smallest class of groups containing. In particular for a group, denotes its isomorphism class.

Examples

The most common examples of classes of groups are:
Given two classes of groups and it is defined the product of classes
This construction allows us to recursively define the power of a class by setting
and
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 ; this group belongs to the class because it has as a subgroup the group which belongs to and furthermore which is in. However has no non-trivial normal cyclic subgroup, so. Then.
However it is straightforward from the definition that for any three classes of groups,, and,

Class maps and closure operations

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:
  1. c is expansive:
  2. c is idempotent:
  3. c is monotonic: If then
Some of the most common examples of closure operations are:
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