Frattini subgroup is normal-monotone
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This fact is an application of the following pivotal fact/result/idea: characteristic of normal implies normal
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Contents
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 be a normal subgroup of a group , where satisfies the property that every proper subgroup is contained in a maximal subgroup. Then, , the Frattini subgroup of , is contained in , the Frattini subgroup of .
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
- Frattini-embedded normal in subgroup and normal implies Frattini-embedded normal: This is the correct generalization, and no longer assumes anything about the group. It is also useful in proving a number of other results.
Proof
Proof outline
The proof uses four facts (for convenience, we denote the group by and normal subgroup by ):
- Characteristic subgroups of normal subgroups are normal: This fact helps us show that the Frattini subgroup of the subgroup , is in fact normal in the whole group
- 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 is a Frattini-embedded normal subgroup inside .
- 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, is a Frattini-embedded normal subgroup inside
- 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 ,a normal subgroup of with the property that any proper subgroup of is contained in a maximal subgroup
To prove:
Proof: The Frattini subgroup is a characteristic subgroup of . Since every characteristic subgroup of a normal subgroup is normal, is a normal subgroup of .
We need to show that is contained in . For this, it suffices to show that is contained in every maximal subgroup of . We prove this by contradiction.
Suppose is a maximal subgroup of not containing . Then, the subgroup generated by and is the whole of . Since is normal in , . From that, it follows that (this is a particular example of the modular property of groups).
Since does not contain , does not contain either and hence is a proper subgroup of . Since every proper subgroup is contained in a maximal subgroup of , there is a maximal subgroup of containing both and . This contradicts the fact that their product is .
Normality of is thus crucial because it guarantees normality of . 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)