Frattini-free group

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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

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 $G$ 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			|FULL LIST, MORE INFO
finite characteristically simple group	finite and characteristicaly simple: nontrivial with no proper nontrivial characteristic subgroup			|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			|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
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.
group in which every Sylow subgroup is elementary abelian	finite abelian group or finite nilpotent group