Frattini subgroup is normal-monotone

From Groupprops
Jump to: navigation, search

Template:Sdf property satisfaction

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic
View other semi-basic facts in group theory
VIEW FACTS USING THIS: directly | directly or indirectly, upto two steps | directly or indirectly, upto three steps|
VIEW: Survey articles about this
This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
View other applications of characteristic of normal implies normal OR Read a survey article on applying characteristic of normal implies normal

Statement

Verbal statement

The Frattini subgroup of any normal subgroup is contained in the Frattini subgroup of the whole group, provided the normal subgroup is a group in which every proper subgroup is contained in a maximal subgroup.

(Note that this group property is always satisfied when the normal subgroup is a finite group, so for finite groups, the Frattini subgroup of a normal subgroup is always contained in the Frattini subgroup of the whole group).

Statement with symbols

Let N be a normal subgroup of a group G, where N satisfies the property that every proper subgroup is contained in a maximal subgroup. Then, \Phi(N), the Frattini subgroup of N, is contained in \Phi(G), the Frattini subgroup of G.

Property-theoretic statement

The subgroup-defining function that sends a group to its Frattini subgroup is a normal-monotone subgroup-defining function (with some assumptions on the nature of the groups).

Definitions

Frattini subgroup

The Frattini subgroup of a group is the intersection of all its maximal subgroups. In other words, an element is in the Frattini subgroup if it is in every maximal subgroup.

Related facts

Proof

Proof outline

The proof uses four facts (for convenience, we denote the group by G and normal subgroup by N):

  1. Characteristic subgroups of normal subgroups are normal: This fact helps us show that the Frattini subgroup of the subgroup N, is in fact normal in the whole group G
  2. For a group where every proper subgroup is contained in a maximal subgroup, the Frattini subgroup is a Frattini-embedded normal subgroup: a normal subgroup whose product with any proper subgroup is proper. Thus \Phi(N) is a Frattini-embedded normal subgroup inside N.
  3. Frattini-embedded normal in subgroup and normal implies Frattini-embedded normal: This allows us to show that the Frattini subgroup of the subgroup, is Frattini-embedded normal inside the whole group. In other words, \Phi(N) is a Frattini-embedded normal subgroup inside G
  4. For any group, any Frattini-embedded normal subgroup is contained inside the Frattini subgroup: This allows us to conclude that the Frattini subgroup of

Hands-on proof

Given: A group G,a normal subgroup N of G with the property that any proper subgroup of N is contained in a maximal subgroup

To prove: \Phi(N) \le \Phi(G)

Proof: The Frattini subgroup \Phi(N) is a characteristic subgroup of N. Since every characteristic subgroup of a normal subgroup is normal, \Phi(N) is a normal subgroup of G.

We need to show that \Phi(N) is contained in \Phi(G). For this, it suffices to show that \Phi(N) is contained in every maximal subgroup of G. We prove this by contradiction.

Suppose M is a maximal subgroup of G not containing \Phi(N). Then, the subgroup generated by M and \Phi(N) is the whole of G. Since \Phi(N) is normal in G, M\Phi(N) = G. From that, it follows that \Phi(N)(M \cap N) = N (this is a particular example of the modular property of groups).

Since M does not contain \Phi(N), M does not contain N either and hence M \cap N is a proper subgroup of N. Since every proper subgroup is contained in a maximal subgroup of N, there is a maximal subgroup of N containing both M \cap N and \Phi(N) . This contradicts the fact that their product is N.

Normality of N is thus crucial because it guarantees normality of \Phi(N). This in turn is crucial in converting a subgroup-generated statement to a product of subgroups statement.

References

Textbook references

  • Abstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347, More info, Exercise 22, Page 199 (Section 6.2)