Frattini-free group

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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

Positive 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
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