GAP:IsTransitiveWithBigGroup

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

This 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 subgroup property and behaves as follows:

• If whenever $H\leq HK\leq G$ are subgroups such that $H$ has property $p$ in $K$ and $K$ has property $p$ in $G$, then $H$ has property $p$ in $G$, the function returns true.
• If GAP is able to find a counterexample, i.e., subgroups $H\leq K\leq G$ such that $K$ has property $p$ in $H$ and $H$ has property $p$ in </math>G</math>, but $K$ does not have property $p$ in $G$, GAP returns false.

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),K -> ForAll(ConjugacyClassSubgroupRepsByProperty(K,p), H -> p(G,H))));
end;;