GAP:IsTransitiveInAllSubgroupsOfGroup

This article is about a GAP function.

This GAP function outputs a Boolean variable, i.e., it returns either true or false. View other GAP functions with Boolean output

Definition

Function type

The function takes as input a group and a subgroup property and outputs a boolean variable (true/false).

Behavior

The function takes as input a group $G$ and a property $p$ and behaves as follows:

• If whenever $H\leq K\leq L\leq G$ such that $H$ has property $p$ in $K$ and $K$ has property $p$ in $L$, $H$ has property $p$ in $L$, then GAP returns true.
• If GAP finds a counterexample, it returns false.

Related functions

• GAP:IsTransitiveWithBigGroup

Method

Code

ConjugacyClassSubgroupReps := function(G)
	return(List(ConjugacyClassesSubgroups(G),Representative));
end;;

ConjugacyClassSubgroupRepsByProperty := function(G,p)
	return(Filtered(ConjugacyClassSubgroupReps(G), H -> p(G,H)));
end;;

IsTransitiveWithBigGroup := function(G,p)
	return(ForAll(ConjugacyClassSubgroupRepsByProperty(G,p),H -> ForAll(ConjugacyClassSubgroupRepsByProperty(H,p), K -> p(G,K))));
end;;

IsTransitiveInAllSubgroupsOfGroup := function(G,p)
	return(ForAll(ConjugacyClassSubgroupReps(G),L -> IsTransitiveWithBigGroup(L,p)));
end;;