# GAP:IsAbnormal

## Contents

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

This function takes as input a pair of two groups, both of which have a common parent group, and returns a Boolean variable (true/false).

### Behavior

Suppose the first group is $G$ and the second group is $H$.

• If, for every $g \in G$, $g$ is in the subgroup generated by $H$ and the conjugate $g^{-1}Hg = H^g$, the function returns true. In particular, if $H \le G$, the function returns true if $H$ is an abnormal subgroup of $G$.
• If there exists $g \in G$ such that $g$ is not in the subgroup generated by $H$ and $H^g$, the function returns false. In particular, if $H$ is a subgroup of $G$ that is not abnormal in $G$, the function returns false.

### Typical use

IsAbnormal(group,subgroup);

## Method

### Code

IsAbnormal := function(G,H)
return(ForAll(Set(G),g-> g in Group(Union(H,H^g))));
end;;