# Frattini-free group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A group is termed **Frattini-free** if its Frattini subgroup (the intersection of all its maximal subgroups) is trivial.

## Examples

### Positive examples

- Any finite simple group or even any finite characteristically simple group, and more generally any simple group or characteristically simple group that has at least one maximal subgroup, is Frattini-free. This is because the Frattini subgroup must be characteristic, so in a characteristically simple group, it must be trivial if it is proper (i.e., if there exist maximal subgroups).
- Any finite group whose order is a square-free number must be Frattini-free. In fact, its maximal subgroups are Hall subgroups that each miss one prime factor, and the intersection of all these is trivial.

### Non-examples

- Any group of prime power order that is not an elementary abelian group is not Frattini-free. The quotient by the Frattini subgroup is the largest possible elementary abelian quotient.
- As a corollary, a finite nilpotent group is Frattini-free only if each of its Sylow subgroups is elementary abelian.
- Any quasisimple group that is not simple (i.e., has nontrivial center) is not Frattini-free: the Frattini subgroup coincides with the nontrivial center.
- The group of rational numbers is not Frattini-free: it has no maximal subgroups, so its Frattini subgroup is the whole group.

## Metaproperties

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

subgroup-closed group property | No | See next column | It is possible to have a group that is Frattini-free, with a subgroup that is not Frattini-free. For instance, take to be symmetric group:S4, and to be a its 2-Sylow subgroup D8 in S4 (we can also take to be Z4 in S4). |

quotient-closed group property | Yes | Suppose is a Frattini-free group and is a normal subgroup of . Then, the quotient group is also Frattini-free. | |

lattice-determined group property | Yes | Whether or not a group is Frattini-free is determined completely by its lattice of subgroups. In other words, if two groups have identical lattices of subgroups, either both are Frattini-free or neither is. |

## Relation with other properties

### Stronger properties

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

finite simple group | finite and simple: nontrivial with no proper nontrivial normal subgroup | K-group, Primitive group|FULL LIST, MORE INFO | ||

finite characteristically simple group | finite and characteristicaly simple: nontrivial with no proper nontrivial characteristic subgroup | Primitive group|FULL LIST, MORE INFO | ||

primitive group | has a core-free maximal subgroup | |FULL LIST, MORE INFO | ||

2-transitive group | has a 2-transitive action on a set | Primitive group|FULL LIST, MORE INFO | ||

K-group | every subgroup is lattice-complemented | |FULL LIST, MORE INFO |

### Conjunction with other properties

Conjunction | Other component of conjunction | Comment |
---|---|---|

finite elementary abelian group | group of prime power order | for a group of prime power order, the Frattini subgroup is the smallest normal subgroup for which the quotient is elementary abelian. |

direct product of elementary abelian Sylow subgroups | finite abelian group or finite nilpotent group |