Frobenius group

A Frobenius group (after Ferdinand Georg Frobenius ) is understood in group theory , a sub-area of algebra , to be a finite group in which a subgroup exists that has the property (where is defined by ). ${\displaystyle G}$ ${\displaystyle 1\subsetneqq H\subsetneqq G}$ ${\displaystyle \forall \,g\in G\setminus H:H^{g}\cap H=1}$ ${\displaystyle H^{g}}$ ${\displaystyle H^{g}:=\{g^{-1}hg|h\in H\}}$

Such a subgroup is then called the Frobenius complement .

An important role in connection with the structure of Frobenius groups plays the so-called Frobenius nucleus, which by

${\displaystyle K:=\left(G\setminus \bigcup _{g\in G}H^{g}\right)\cup \left\{1\right\}}$

is defined. More precisely, one speaks of the Frobenius nucleus in relation to the Frobenius complement . ${\displaystyle G}$ ${\displaystyle H}$

Structure of Frobenius groups

Frobenius' theorem about Frobenius groups says that holds. In fact, it is and is considered a semi-direct product . Furthermore, hallsch are in and for every two Frobenius complements of there is one with the property or . ${\displaystyle K\leq G}$ ${\displaystyle K\triangleleft G}$ ${\displaystyle G=HK}$ ${\displaystyle H,K}$ ${\displaystyle G}$ ${\displaystyle A,B}$ ${\displaystyle G}$ ${\displaystyle g\in G}$ ${\displaystyle A^{g}\leq B}$ ${\displaystyle B^{g}\leq A}$

Structure without representation theory

The above-mentioned theorem of Frobenius can only be proven today with means of representation theory. From today's point of view, it is therefore interesting to provide evidence on an exclusively group-theoretical level. Sometimes this has not yet been successful, but partial proofs have been provided that demonstrate the subgroup property of under stronger conditions. If one of the following properties applies, then ${\displaystyle K}$ ${\displaystyle K\leq G}$

• ${\displaystyle \left|H\right|\equiv 0\mod 2}$
• ${\displaystyle H}$ is resolvable
• ${\displaystyle \exists \,1\neq D\triangleleft G:D\subseteq K}$
• ${\displaystyle \exists \,1\neq D\leq G:H\leq N_{G}(D)}$
• ${\displaystyle \exists \,h\in H^{\#}\,\forall \,g\in G:\left\langle h,h^{g}\right\rangle }$ dissolvable
• ${\displaystyle G}$is not easy
• ${\displaystyle \exists \,1\neq D\leq G:D\subseteq K\;\wedge \;\pi (D)\neq \left\{2\right\}\;\wedge \;N_{G}(D)\cap K\neq N_{G}(D)}$
• ${\displaystyle \exists \,L\leq G\,\exists \,l\in L\cap K:L}$ dissolvable ${\displaystyle \;\wedge \;L\cap K\neq 1\;\wedge \;L\cap K\neq L\;\wedge \;o(l)\equiv 1\mod 2}$

All of these results can be achieved without the aid of representation theory. Note that the first two statements, with the help of Feit & Thompson's theorem, already provide that there is always a subgroup. Feit and Thompson's theorem says that groups of odd order are always solvable. If the order of a Frobenius group is even, then point 1 yields , otherwise the order is odd, so that according to Feit-Thompson the group can be resolved, so that point 2 results. ${\displaystyle K}$ ${\displaystyle K\leq G}$ ${\displaystyle K\leq G}$

However, this sentence is also proven with the aid of representation theory, so that it cannot be used in this context.

Examples

• The symmetrical group of degree 3 is a Frobenius group. Three different Frobenius complements come into question, namely for each of the three transpositions of the group. The Frobenius nucleus is then the alternating group . ${\displaystyle \left\langle \tau \right\rangle }$ ${\displaystyle \tau \in \left\{(12),(13),(23)\right\}}$ ${\displaystyle A_{3}}$
• The group of invertible - triangle matrices with determinant 1 over a finite field with is a Frobenius group. The subgroup of diagonal matrices is the Frobenius complement and the group of strict triangular matrices ( only ones in the main diagonal ) is the Frobenius kernel . ${\displaystyle 2\times 2}$ ${\displaystyle \left|K\right|\geq 3}$

swell

• Paul J. Flavell: A Note on Frobenius groups , Journal of Algebra 228, p. 367 ff. ( Pdf file )
• Nathan Jacobson: Basic Algebra II . P. 317 1980 WH Freeman and Company ISBN 071671079X .