# GAP:IsSelfCentralizing

From Groupprops

## Contents

## Definition

### Function type

The function takes in two arguments, both of them groups, and outputs a Boolean variable.

### Behavior

- The function returns
`true`if both groups are subgroups of a common big group, and if the centralizer of the second group in the first is also contained in the second subgroup. - The function returns
`false`if the two groups are subgroups of a common big group, and if the centralizer of the second group in the first is*not*contained in the second group. - If the two groups are not contained in a common big group, the function returns a
`NoMethodFound`error.

### Typical usage

The function is typically invoked as follows:

IsSelfCentralizing(group,subgroup);

The function returns `true` if the subgroup is a self-centralizing subgroup of the whole group and `false` otherwise.

## Related functions

## Method

### Code

IsSelfCentralizing := function(G,H) return (Centralizer(G,H) <= H); end;;