Frattini's argument

The argument Frattini , in short Frattini argument , one after the Italian mathematician Giovanni Frattini called final way from the mathematical branch of group theory . It enables a finite group to be written as a complex product of two subgroups under certain circumstances .

Definitions

We use the power notation for conjugation , that is, are and elements of a group , so we write and for a subset . As is well known, normal subgroups are precisely those subgroups that apply to all and denote the normalizer of in . For a prime number , a p -Sylow group is a p subgroup of maximum order . ${\displaystyle x}$ ${\displaystyle y}$ ${\displaystyle G}$ ${\displaystyle x^{y}:=y^{-1}xy}$ ${\displaystyle M^{y}:=\{y^{-1}xy|\,x\in M\}}$ ${\displaystyle M}$ ${\displaystyle N}$ ${\displaystyle N^{y}=N}$ ${\displaystyle y\in G}$ ${\displaystyle N_{G}(M):=\{y\in G|\,M^{y}=M\}}$ ${\displaystyle M}$ ${\displaystyle G}$ ${\displaystyle p}$

The Frattini argument

If the group is a normal divisor and a p -Sylow group of , then applies . ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle P}$ ${\displaystyle N}$ ${\displaystyle G=N_{G}(P)\cdot N}$

If, namely , then is , also p -Sylow group in . The Sylow theorems for result in and in are conjugated, that is, there is a with . It follows , so and with it . Since was arbitrary, the assertion follows. ${\displaystyle x\in G}$ ${\displaystyle P^{x}\subset N^{x}=N}$ ${\displaystyle P^{x}}$ ${\displaystyle N}$ ${\displaystyle N}$ ${\displaystyle P}$ ${\displaystyle P^{x}}$ ${\displaystyle N}$ ${\displaystyle y\in N}$ ${\displaystyle P^{y}=P^{x}}$ ${\displaystyle P^{xy^{-1}}=P}$ ${\displaystyle xy^{-1}\in N_{G}(P)}$ ${\displaystyle x\in N_{G}(P)\cdot y\subset N_{G}(P)\cdot N}$ ${\displaystyle x\in G}$

Further definitions

A reasoning closely related to the above Frattini argument also exists for the operation of a group G on a set Ω . An operation is transitive if for any two elements a are with . For is the so-called stabilizer group in . Also note that each of its subgroups also operates on. With these terms, the following situation, also known as the Frattini argument, applies: ${\displaystyle \omega _{1},\omega _{2}\in \Omega }$ ${\displaystyle x\in G}$ ${\displaystyle \omega _{1}^{x}=\omega _{2}}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle G_{\omega }:=\{x\in G|\,\omega ^{x}=\omega \}}$ ${\displaystyle \omega }$ ${\displaystyle G}$ ${\displaystyle \Omega }$

The Frattini argument for operations

The group was operating on , is a subset of , and in limited operation to be transitive. Then applies to each . ${\displaystyle G}$ ${\displaystyle \Omega }$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle N}$ ${\displaystyle \Omega }$ ${\displaystyle G=G_{\omega }\cdot N}$ ${\displaystyle \omega \in \Omega }$

The proof of this statement is an elementary variant of the conclusion presented above. Namely, is and , so is and because of the presupposed transitivity of there is with , that is , so and finally . There and were arbitrary, the assertion follows. ${\displaystyle x\in G}$ ${\displaystyle \omega \in \Omega }$ ${\displaystyle \omega ^{x}\in \Omega }$ ${\displaystyle N}$ ${\displaystyle y\in N}$ ${\displaystyle \omega ^{y}=\omega ^{x}}$ ${\displaystyle \omega ^{xy^{-1}}=\omega }$ ${\displaystyle xy^{-1}\in G_{\omega }}$ ${\displaystyle x\in G_{\omega }\cdot y\subset G_{\omega }\cdot N}$ ${\displaystyle x\in G}$ ${\displaystyle \omega \in \Omega }$

The Frattini argument for normal divisors can be reduced to the Frattini argument for operations. If the set of p -Sylow groups is , then operates by conjugation on and the operation restricted to is transitive according to Sylow's theorems. The stabilizer group is closed for each . So the Frattini argument for operations is . ${\displaystyle \Omega }$ ${\displaystyle N}$ ${\displaystyle G}$ ${\displaystyle \Omega }$ ${\displaystyle N}$ ${\displaystyle P\in \Omega }$ ${\displaystyle N_{G}(P)}$ ${\displaystyle P}$ ${\displaystyle G=N_{G}(P)\cdot N}$

Applications

• The first application, going back to Frattini himself, consists in the proof that the so-called Frattini group of a finite group is nilpotent .
• If a p -Sylow group is a finite group , then . To do this, apply the Frattini argument to the group , which contains the normal divisor. ${\displaystyle P}$ ${\displaystyle G}$ ${\displaystyle N_{G}(N_{G}(P))=N_{G}(P)}$ ${\displaystyle N_{G}(N_{G}(P))}$ ${\displaystyle N_{G}(P)}$

Individual evidence

1. ^ H. Kurzweil, B. Stellmacher: Theory of finite groups , Springer-Verlag (1998), ISBN 3-540-60331-X , section 3.2.7
2. ^ Derek JS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , section 5.2.14
3. ↑ H. Kurzweil, B. Stellmacher: Theory of finite groups , Springer-Verlag (1998), ISBN 3-540-60331-X , section 3.1.4
4. ↑ G. Frattini: Intorno alla generazione dei gruppi di operazioni , Rome. Acc. L. Rend. (4) I. 281-285, 455-457, 1885.