GAP:IsComplementedNormalSubgroup

This article is about a GAP function.

This GAP function takes as input two groups, both having a common parent group. See other GAP functions like this.
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 two groups, both of which are subgroups of a common big group, and outputs a boolean (true/false).

Behavior

The function behaves as follows:

• If the second group is not a subgroup of the first, but they are both subgroups of a common big group, the function returns false.
• If the second group is a subgroup of the first, but is not a complemented normal subgroup, the function returns false.
• If the second group is a complemented normal subgroup of the first, the function returns true.
• If the arguments are not groups, or if they do not live inside a common parent group, GAP returns a NoMethodFound error.

Method

Code

IsComplementedNormalSubgroup := function(G,H)
        return(IsSubgroup(G,H) and IsNormal(G,H) and ForAny(List(ConjugacyClassesSubgroups(G),Representative),L -> Order(H) * Order(L) = Order(G) and IsTrivial(Intersection(H,L))));
end;;