# P-constrained group

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

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## Contents

## Definition

### Definition for a general finite group

Let be a finite group and be a prime number. We say that is -constrained if the following is true for one (and hence, any) -Sylow subgroup of :

.

Here, denotes the centralizer of in . is the second member of the lower pi-series for .

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

This is the same as the previous definition, restricted to p'-core-free groups.

Let be a finite group and be a prime number. Suppose further that the p'-core of is trivial, i.e., is the trivial group. Equivalently, every nontrivial normal subgroup of has order divisible by . Then, we say that is -constrained if its p-core is a self-centralizing subgroup, i.e.,:

### Equivalence of definitions and its significance

`Further information: equivalence of definitions of p-constrained group`

It turns out that, from the above definitions:

is -constrained is -constrained.

This allows us to define -constraint for arbitrary finite groups in terms of -constraint for -core-free finite groups.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

strongly p-solvable group | ||||

p-solvable group | p-solvable implies p-constrained | p-constrained not implies p-solvable | ||

finite solvable group | (via p-solvable) | (via p-solvable) | ||

p-nilpotent group | (via p-solvable) | (via p-solvable) | ||

finite nilpotent group | (via finite solvable) | (via finite solvable) |

### Incomparable properties

Property | Meaning | Proof of one non-implication | Proof of other non-implication |
---|---|---|---|

p-stable group | p-constrained not implies p-stable | p-stable not implies p-constrained | |

group of Glauberman type for a prime | p-constrained not implies Glauberman type | Glauberman type not implies p-constrained |

## Metaproperties

Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|

subgroup-closed group property | No | p-constraint is not subgroup-closed | It is possible to have a finite group , a subgroup , and a prime number such that is -constrained and is not. |

quotient-closed group property | No | p-constraint is not quotient-closed | It is possible to have a finite group , a normal subgroup , and a prime number such that is -constrained and is not. |