Class of groups explained

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

ak{X}

is a collection of groups such that if

G\inak{X}

and

G\congH

then

H\inak{X}

. Groups in the class

ak{X}~

are referred to as

ak{X}

-groups.

For a set of groups

ak{I}

, we denote by

(ak{I})

the smallest class of groups containing

ak{I}

. In particular for a group

(G)

denotes its isomorphism class.

Examples

The most common examples of classes of groups are:

\emptyset

: the empty class of groups

ak{C}~

: the class of cyclic groups

ak{A}~

: the class of abelian groups

ak{U}~

: the class of finite supersolvable groups

ak{N}~

: the class of nilpotent groups

ak{S}~

: the class of finite solvable groups

ak{I}~

: the class of finite simple groups

ak{F}~

: the class of finite groups

ak{G}~

: the class of all groups

Product of classes of groups

Given two classes of groups

ak{X}

and

ak{Y}

it is defined the product of classes

ak{X}ak{Y}=(G\midGhasanormalsubgroupN\inak{X}withG/N\inak{Y}).

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

ak{X}⁰=(1)

and

ak{X}ⁿ=ak{X}^n-1ak{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

(ak{C}ak{C})ak{C}

because it has as a subgroup the group

V₄

, which belongs to

ak{C}ak{C}

, and furthermore

A_4/V_4\congC₃

, which is in

ak{C}

. However

A₄

has no non-trivial normal cyclic subgroup, so

A_{4\not\inak{C}(ak{C}ak{C})}

. Then

ak{C}(ak{C}ak{C})\not=(ak{C}ak{C})ak{C}

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

ak{X}

ak{Y}

, and

ak{Z}

ak{X}(ak{Y}ak{Z})\subseteq(ak{X}ak{Y})ak{Z}

Class maps and closure operations

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

ak{X}

to another class of groups

cak{X}

. A class map is said to be a closure operation if it satisfies the next properties:

c is expansive:

ak{X}\subseteqcak{X}

c is idempotent:

cak{X}=c(cak{X})

c is monotonic: If

ak{X}\subseteqak{Y}

then

cak{X}\subseteqcak{Y}

Some of the most common examples of closure operations are:

Sak{X}=(G\midG\leqH, H\inak{X})

Qak{X}=(G\midexistsH\inak{X}andanepimorphismfromHtoG)

N_0ak{X}=(G\midexistsK_{i (i=1, … ,r)subnormalinGwithK}_i\inak{X}andG=\langleK_{1, … ,K}_r\rangle)

R_0ak{X}=(G\midexistsN_{i (i=1, … ,r)normalinGwithG/N}_i\in

	rNi=1)
ak{X}andcap\limits
	i=1

S_nak{X}=(G\midGissubnormalinHforsomeH\inak{X})

Class of groups explained

Definition

Examples

Product of classes of groups

Class maps and closure operations

See also