# Group of Glauberman type for a prime

The article defines a property of groups, where the definition may be in terms of a particular prime that serves as parameter

View other prime-parametrized group properties | View other group properties

## Contents

## Definition

### Definition for a general finite group

Suppose is a finite group and is a prime number. We say that is of **Glauberman type** with respect to if the ZJ-functor is a characteristic p-functor whose normalizer generates whole group with p'-core. In the discussion below, denotes the subgroup obtained by applying the ZJ-functor to . The ZJ-functor is defined as the center of the Thompson subgroup , which in turn is defined as the join of abelian subgroups of maximum order.

Explicitly, the following equivalent conditions are satisfied:

- For one (and hence every) -Sylow subgroup of , : Here, denotes the -core of ,
- For one (and hence every) -Sylow subgroup of , the image of in the quotient is a normal subgroup of .
- For one (and hence every) -Sylow subgroup of , is a normal subgroup of .
- For one (and hence every) -Sylow subgroup of , is a characteristic subgroup of .

### Definition for p'-core-free finite group

Suppose is a finite group and is a prime number. Suppose the p'-core is trivial, i.e., has no nontrivial normal subgroup of order not divisible by . Then, is termed a **group of Glauberman type** for if it satisfies the following equivalent conditions:

- For one (and hence every) -Sylow subgroup , is a normal subgroup of .
- For one (and hence every) -Sylow subgroup , is a characteristic subgroup of .

### Equivalence of definitions and its significance

`Further information: Equivalence of definitions of group of Glauberman type for a prime`

It turns out that, for a finite group and prime number :

is a group of Glauberman type for is a group of Glauberman type for

This can be used to provide an alternative definition of group of Glauberman type.

## Relation with other properties

### Stronger properties

- p-nilpotent group
- strongly p-solvable group (for an odd prime ):
`For full proof, refer: strongly p-solvable implies Glauberman type for odd p` - Group that is both p-stable and p-constrained for an odd prime :
`For full proof, refer: p-constrained and p-stable implies Glauberman type for odd p`

### Weaker properties

- Group in which the ZJ-functor controls fusion for a prime:
`For full proof, refer: Glauberman type implies ZJ-functor controls fusion`

## Examples

### p'-core-free examples

In these example groups, is normal in for one (and hence every) -Sylow subgroup of .

Group | Order | Prime of interest | -Sylow subgroup | Comment | |
---|---|---|---|---|---|

symmetric group:S3 | 6 | 3 | A3 in S3 | A3 in S3 | Here, and it itself is normal in . |

alternating group:A4 | 12 | 2 | V4 in A4 | V4 in A4 | Here, and it itself is normal in . |

On the other hand, symmetric group:S4 is *not* a group of Glauberman type for the prime 2.