# Collection of groups satisfying a weak normal replacement condition

From Groupprops

## Contents

## Statement

Suppose is a collection of finite -groups, i.e., groups of prime power order where the underlying prime is . We say that satisfies a **weak normal replacement condition** if whenever a finite -group contains a subgroup isomorphic to an element of , also contains a normal subgroup isomorphic to an element of .

## Relation with other properties

### Stronger properties

- Collection of groups satisfying a universal congruence condition
- Collection of groups satisfying a strong normal replacement condition

## Examples/facts

### Satisfaction

Collection of groups of order | Condition on prime | Condition on | Proof |
---|---|---|---|

Elementary abelian group of order | Odd | Jonah-Konvisser congruence condition on number of elementary abelian subgroups of small prime power order for odd prime | |

Elementary abelian group of order | Odd | Elementary abelian-to-normal replacement theorem for large primes | |

Abelian groups of order | Odd | Jonah-Konvisser congruence condition on number of abelian subgroups of small prime power order for odd prime | |

Abelian groups of order , exponent dividing | Odd | Congruence condition on number of abelian subgroups of small prime power order and bounded exponent for odd prime | |

Abelian groups of order , exponent dividing | Any | Glauberman's abelian-to-normal replacement theorem for bounded exponent and half of prime plus one | |

Groups of order , exponent | -- | Mann's replacement theorem for subgroups of prime exponent |

### Dissatisfaction

Collection of groups of order | Condition on prime | Condition on | Proof |
---|---|---|---|

Klein four-group | Elementary abelian-to-normal replacement fails for Klein four-group | ||

Elementary abelian group of order | elementary abelian-to-normal replacement fails for half of prime plus nine for prime greater than five | ||

Abelian groups of order | abelian-to-normal replacement fails for half of prime plus nine for prime greater than five | ||

Groups of order , exponent | all |

### Threshold values

This lists threshold values of : the largest value of for which the collection of -groups of order satisfying the stated condition satisfies a weak normal replacement condition. The nature of all these is such that the weak normal replacement condition is satisfied for all smaller but for no larger . The *between and * below means that the minimum known value is and the maximum known value is .

Collection of groups | ||||||
---|---|---|---|---|---|---|

Abelian groups of order | between 4 and 5 | between 5 and 13 | between 5 and 6 | between 5 and 7 | between 6 and 9 | between and |

Abelian groups of order , exponent dividing | between 2 and 5 | between 5 and 13 | between 5 and 6 | between 5 and 7 | between 6 and 9 | between and |

Elementary abelian group of order | 1 | between 5 and 13 (?) | between 5 and 6 (?) | between 5 and 7 | between 6 and 9 | between and |

Groups of exponent , order | 1 | at least 2 | at least 4 | at least 6 |