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)