A class of groups ${\displaystyle {\mathfrak {X}}}$  is a collection of groups such that if ${\displaystyle G\in {\mathfrak {X}}}$  and ${\displaystyle G\cong H}$  then ${\displaystyle H\in {\mathfrak {X}}}$ . Groups in the class ${\displaystyle {\mathfrak {X}}~}$  are referred to as ${\displaystyle {\mathfrak {X}}}$ -groups.

For a set of groups ${\displaystyle {\mathfrak {I}}}$ , we denote by ${\displaystyle ({\mathfrak {I}})}$  the smallest class of groups containing ${\displaystyle {\mathfrak {I}}}$ . In particular for a group ${\displaystyle G}$ , ${\displaystyle (G)}$  denotes its isomorphism class.

The most common examples of classes of groups are:

• ${\displaystyle \emptyset }$ : the empty class of groups
• ${\displaystyle {\mathfrak {C}}~}$ : the class of cyclic groups
• ${\displaystyle {\mathfrak {A}}~}$ : the class of abelian groups
• ${\displaystyle {\mathfrak {U}}~}$ : the class of finite supersolvable groups
• ${\displaystyle {\mathfrak {N}}~}$ : the class of nilpotent groups
• ${\displaystyle {\mathfrak {S}}~}$ : the class of finite solvable groups
• ${\displaystyle {\mathfrak {I}}~}$ : the class of finite simple groups
• ${\displaystyle {\mathfrak {F}}~}$ : the class of finite groups
• ${\displaystyle {\mathfrak {G}}~}$ : the class of all groups

Given two classes of groups ${\displaystyle {\mathfrak {X}}}$  and ${\displaystyle {\mathfrak {Y}}}$  it is defined the product of classes

${\displaystyle {\mathfrak {X}}{\mathfrak {Y}}=(G\mid G{\text{ has a normal subgroup }}N\in {\mathfrak {X}}{\text{ with }}G/N\in {\mathfrak {Y}}).}$ 

This construction allows us to recursively define the power of a class by setting

${\displaystyle {\mathfrak {X}}^{0}=(1)}$  and ${\displaystyle {\mathfrak {X}}^{n}={\mathfrak {X}}^{n-1}{\mathfrak {X}}.}$ 

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 ${\displaystyle ({\mathfrak {C}}{\mathfrak {C}}){\mathfrak {C}}}$  because it has as a subgroup the group ${\displaystyle V_{4}}$ , which belongs to ${\displaystyle {\mathfrak {C}}{\mathfrak {C}}}$ , and furthermore ${\displaystyle A_{4}/V_{4}\cong C_{3}}$ , which is in ${\displaystyle {\mathfrak {C}}}$ . However ${\displaystyle A_{4}}$  has no non-trivial normal cyclic subgroup, so ${\displaystyle A_{4}\not \in {\mathfrak {C}}({\mathfrak {C}}{\mathfrak {C}})}$ . Then ${\displaystyle {\mathfrak {C}}({\mathfrak {C}}{\mathfrak {C}})\not =({\mathfrak {C}}{\mathfrak {C}}){\mathfrak {C}}}$ .

However it is straightforward from the definition that for any three classes of groups ${\displaystyle {\mathfrak {X}}}$ , ${\displaystyle {\mathfrak {Y}}}$ , and ${\displaystyle {\mathfrak {Z}}}$ ,

${\displaystyle {\mathfrak {X}}({\mathfrak {Y}}{\mathfrak {Z}})\subseteq ({\mathfrak {X}}{\mathfrak {Y}}){\mathfrak {Z}}}$ 

A class map c is a map which assigns a class of groups ${\displaystyle {\mathfrak {X}}}$  to another class of groups ${\displaystyle c{\mathfrak {X}}}$ . A class map is said to be a closure operation if it satisfies the next properties:

1. c is expansive: ${\displaystyle {\mathfrak {X}}\subseteq c{\mathfrak {X}}}$ 
2. c is idempotent: ${\displaystyle c{\mathfrak {X}}=c(c{\mathfrak {X}})}$ 
3. c is monotonic: If ${\displaystyle {\mathfrak {X}}\subseteq {\mathfrak {Y}}}$  then ${\displaystyle c{\mathfrak {X}}\subseteq c{\mathfrak {Y}}}$ 

Some of the most common examples of closure operations are:

• ${\displaystyle S{\mathfrak {X}}=(G\mid G\leq H,\ H\in {\mathfrak {X}})}$ 
• ${\displaystyle Q{\mathfrak {X}}=(G\mid {\text{exists }}H\in {\mathfrak {X}}{\text{ and an epimorphism from }}H{\text{ to }}G)}$ 
• ${\displaystyle N_{0}{\mathfrak {X}}=(G\mid {\text{ exists }}K_{i}\ (i=1,\cdots ,r){\text{ subnormal in }}G{\text{ with }}K_{i}\in {\mathfrak {X}}{\text{ and }}G=\langle K_{1},\cdots ,K_{r}\rangle )}$ 
• ${\displaystyle R_{0}{\mathfrak {X}}=(G\mid {\text{ exists }}N_{i}\ (i=1,\cdots ,r){\text{ normal in }}G{\text{ with }}G/N_{i}\in {\mathfrak {X}}{\text{ and }}\bigcap \limits _{i=1}^{r}Ni=1)}$ 
• ${\displaystyle S_{n}{\mathfrak {X}}=(G\mid G{\text{ is subnormal in }}H{\text{ for some }}H\in {\mathfrak {X}})}$ 

• Ballester-Bolinches, Adolfo; Ezquerro, Luis M. (2006), Classes of finite groups, Mathematics and Its Applications (Springer), vol. 584, Berlin, New York: Springer-Verlag, ISBN 978-1-4020-4718-3, MR 2241927
• Doerk, Klaus; Hawkes, Trevor (1992), Finite soluble groups, de Gruyter Expositions in Mathematics, vol. 4, Berlin: Walter de Gruyter & Co., ISBN 978-3-11-012892-5, MR 1169099

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