# Equivalence of normality and characteristicity conditions for isomorph-free p-functor

## Contents

## Statement

Let be a prime number. Suppose is a characteristic p-functor that always returns an isomorph-free subgroup of its input group. Then, the following are equivalent for a finite group :

- For every pair of -Sylow subgroups of , .
- For every pair of -Sylow subgroups of , .
- Each of these:
- There exists a -Sylow subgroup of such that where is the p-core of .
- For every -Sylow subgroup of , where is the -core of .

- Each of these:
- There exists a -Sylow subgroup of such that is a characteristic subgroup of .
- For every -Sylow subgroup of , is a characteristic subgroup of .

- Each of these:
- There exists a -Sylow subgroup of such that is a normal subgroup of .
- For every -Sylow subgroup of , is a normal subgroup of .

## Related facts

- Equivalence of normality and characteristicity conditions for conjugacy functor: This is more general, because any isomorph-free p-functor is weakly closed, but the converse need not hold.
- Equivalence of definitions of weakly closed conjugacy functor: This is a somewhat more powerful version of the statement, albeit it has different hypotheses and conclusions. It can really be thought of as a
*local*version.

## Facts used

## Proof

Fact (1) shows that the *For every* versions are equivalent to the *there exists* versions. This proves the equivalence of the two versions of (3), the two versions of (4), and the two versions of (5). The remaining directions are each individually quite easy and are summarized below.

We prove the equivalence of (1) with (2). Within this the (2) implies (1) direction requires the use of isomorph-free. We prove the equivalence of (2) with (3). This is straightforward. We then cyclically prove the equivalence of (3), (4) and (5). Of this cyclic proof, the (3) implies (4) part uses isomorph-free via Facts (2) and (3).

From | To | Facts used | Given data used | Explanation |
---|---|---|---|---|

Condition (1) | Condition (2) | -- | -- | Immediate from the fact that on account of being a characteristic -functor. |

Condition (2) | Condition (1) | is isomorph-free | If , then we know that are isomorphic subgroups of . By the isomorph-free condition, we must have . | |

Condition (2) | Condition (3) | -- | -- | If for all -Sylow subgroups , we have . |

Condition (3) | Condition (2) | -- | -- | and for all -Sylow subgroups forces . |

Condition (3) | Condition (4) | Facts (2),(3) | is isomorph-free | We have . By Fact (2), is characteristic in . Separately, is characteristic in . Fact (3) now gives that is characteristic in . |

Condition (4) | Condition (5) | Fact (4) | -- | Fact-direct |

Condition (5) | Condition (3) | Fact (5) | By definition of p-core, being normal forces . |