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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- 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.
- 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.
|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
|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|