GAP:IsRetract

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

This GAP function is not in-built: you need to copy the code on this page to define the function.

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 behavior is as follows:

• If the second group is not a subgroup of the first group, the function returns false.
• If the second group is a retract of the first group, i.e., it possesses a normal complement, the function returns true.
• If the second group is a subgroup of the first, but is not a retract, the function returns false.

Method

Code

This code works only for finite groups:

IsRetract := function(G,H)
        return(IsSubgroup(G,H) and ForAny(NormalSubgroups(G),L -> Order(L) * Order(H) = Order(G) and IsTrivial(Intersection(H,L))));
end;;