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".
Definition.
A class of groups formula_1 is a collection of groups such that if formula_2 and formula_3 then formula_4. Groups in the class formula_1 are referred to as formula_6-groups.
For a set of groups formula_7, we denote by formula_8 the smallest class of groups containing formula_9. In particular for a group formula_10, formula_11 denotes its isomorphism class.
Examples.
The most common examples of classes of groups are:
Product of classes of groups.
Given two classes of groups formula_1 and formula_22 it is defined the product of classes
formula_23
This construction allows us to recursively define the power of a class by setting
formula_24 and formula_25
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 formula_26 because it has as a subgroup the group formula_27 which belongs to formula_28 and furthermore formula_29 which is in formula_13. However formula_31 has no non-trivial normal cyclic subgroup, so formula_32. Then formula_33.
However it is straightforward from the definition that for any three classes of groups formula_6, formula_35, and formula_36,
formula_37
Class maps and closure operations.
A class map c is a map which assigns a class of groups formula_6 to another class of groups formula_39. 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:
See also.
Formation