# GAP:IsAbnormal

From Groupprops

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

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 and the second group is .

- If, for every , is in the subgroup generated by and the conjugate , the function returns
`true`. In particular, if , the function returns`true`if is an abnormal subgroup of . - If there exists such that is not in the subgroup generated by and , the function returns
`false`. In particular, if is a subgroup of that is not abnormal in , 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;;