Frattini subgroup is nilpotent in finite: Difference between revisions

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Statement

Verbal statement

The Frattini subgroup of any finite group is nilpotent.

Symbolic statement

Let $G$ be a finite group and $Φ (G)$ denote the intersection of all maximal subgroups of $G$ (the so-called Frattini subgroup of $G$ ).

Definitions used

Frattini subgroup

Further information: Frattini subgroup

The Frattini subgroup of a (here, finite) group is the intersection of all its maximal subgroups.

Finite nilpotent group

Further information: Finite nilpotent group A finite group is nilpotent if every Sylow subgroup of it is normal.

Generalizations

The result generalizes in two important respects. First, we can prove considerably stronger results about Frattini subgroups for finite groups, and more generally, for groups in which every proper subgroup is contained in a maximal subgroup. Second, we can generalize to proving the results about any Frattini-embedded normal subgroup of an arbitrary group.

For instance:

Proof

Given: A finite group $G$ , and $Φ (G)$ is the Frattini subgroup

To prove: For any Sylow subgroup $P$ of $Φ (G)$ , $P$ is normal in $Φ (G)$ .

Proof: In fact, we shall show that $P$ is normal in $G$ .

Here's the idea. By applying Frattini's argument and the fact that $Φ (G) ◃ G$ , we have $Φ (G) N_{G} (P) = G$ . Now if $N_{G} (P) \neq G$ , it is contained in a maximal subgroup $M$ of $G$ . Since $Φ (G)$ is contained in every maximal subgroup, $Φ (G) N_{G} (P) \leq M$ , leading to a contradiction.

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 25, Page 199 (Section 6.2)

@@ Line 16: / Line 16: @@
 ===Frattini subgroup===
+{{further|[[Frattini subgroup]]}}
 The [[Frattini subgroup]] of a (here, finite) group is the intersection of all its [[maximal subgroup]]s.
-===Nilpotent group===
+===Finite nilpotent group===
+{{further|[[Finite nilpotent group]]}}
 A [[finite group]] is nilpotent if every [[Sylow subgroup]] of it is [[normal subgroup|normal]].
 ==Generalizations==
-There is a somewhat more general version of this result: the Frattini subgroup of a finite group (or more generally, a group where every subgroup is contained in a [[maximal subgroup]]) has the property of being an [[ACIC-group]]: any [[automorph-conjugate subgroup]] in it is [[characteristic subgroup|characteristic]]. {{proofat|[[Frattini subgroup is ACIC]]}}.
+The result generalizes in two important respects. First, we can prove considerably stronger results about Frattini subgroups for finite groups, and more generally, for groups in which every proper subgroup is contained in a maximal subgroup. Second, we can generalize to proving the results about any [[Frattini-embedded normal subgroup]] of an arbitrary group.
+For instance:
+* [[Frattini-embedded normal-realizable implies inner-in-automorphism-Frattini]]
+* [[Frattini-embedded normal-realizable implies ACIC]]
 ==Proof==
-We are given a finite group <math>G</math>, and <math>\Phi(G)</math> is the [[Frattini subgroup]]. We need to show that for any [[Sylow subgroup]] <math>P</math> of <math>\Phi(G)</math>, <math>P</math> is normal in <math>\Phi(G)</math>.
+''Given'': A finite group <math>G</math>, and <math>\Phi(G)</math> is the [[Frattini subgroup]]
+''To prove'': For any [[Sylow subgroup]] <math>P</math> of <math>\Phi(G)</math>, <math>P</math> is normal in <math>\Phi(G)</math>.
-In fact, we shall show that <math>P</math> is normal in <math>G</math>.
+''Proof'': In fact, we shall show that <math>P</math> is normal in <math>G</math>.
 Here's the idea. By applying [[Frattini's argument]] and the fact that <math>\Phi(G) \triangleleft G</math>, we have <math>\Phi(G)N_G(P) = G</math>. Now if <math>N_G(P) \ne G</math>, it is contained in a maximal subgroup <math>M</math> of <math>G</math>. Since <math>\Phi(G)</math> is contained in ''every'' maximal subgroup, <math>\Phi(G)N_G(P) \le M</math>, leading to a contradiction.
+==References==
+===Textbook references===
+* {{booklink-stated|DummitFoote}}, Exercise 25, Page 199 (Section 6.2)