# Frattini-free group

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

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

## Examples

### 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 $G$ that is Frattini-free, with a subgroup $H$ that is not Frattini-free. For instance, take $G$ to be symmetric group:S4, and $H$ to be a its 2-Sylow subgroup D8 in S4 (we can also take $H$ to be Z4 in S4).
quotient-closed group property Yes Suppose $G$ is a Frattini-free group and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ 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 $G_1, G_2$ 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