Class of groups

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

Formation