Minor class graph

The sub-class graph is a graph-theoretic aid of group theory . Through it, some group theoretical facts can be formulated clearly and simply. In the past some proofs could be simplified and greatly reduced by him.

definition

Be a group , and be subgroups of . Let be the graph with vertex set , all secondary classes after the , and the edge set . Then the subclass graph is called after the . ${\displaystyle G}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle P_{1},...,P_{n}}$ ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle X(\Gamma ):=\{P_{i}x|1\leq i\leq n,x\in G\}}$ ${\displaystyle P_{i}}$ ${\displaystyle K(\Gamma ):=\{\{\alpha ,\beta \}|\alpha \neq \beta \in X(\Gamma ),\alpha \cap \beta \neq \emptyset \}}$ ${\displaystyle \Gamma }$ ${\displaystyle P_{i}}$

Properties of Γ

${\displaystyle G}$ operates by law multiplication on and . One often speaks of the operation of on , whereby it can be seen from the context which of the two operations is meant. In most cases we are talking about the operation on the set of vertices. ${\displaystyle X(\Gamma )}$ ${\displaystyle K(\Gamma )}$ ${\displaystyle G}$ ${\displaystyle \Gamma }$

The operation from to is divided into orbits, each of which represents a representative of these orbits. (In particular, n-partit is with partitions ). ${\displaystyle G}$ ${\displaystyle \Gamma }$ ${\displaystyle n}$ ${\displaystyle P_{1},...,P_{n}}$ ${\displaystyle \Gamma }$ ${\displaystyle \{P_{i}x|x\in G\}}$

Designations

Be . Then denote the trajectory from below and the stabilizer from in . With the amount of neighbors had called. ${\displaystyle \alpha \in \Gamma }$ ${\displaystyle \alpha ^{G}}$ ${\displaystyle \alpha }$ ${\displaystyle G}$ ${\displaystyle G_{\alpha }}$ ${\displaystyle \alpha }$ ${\displaystyle G}$ ${\displaystyle \Delta (\alpha )}$ ${\displaystyle \alpha }$

Simple properties

Be . Then: ${\displaystyle \alpha \in \Gamma }$

• ${\displaystyle G_{\alpha }}$is conjugated to one of the . More precisely: is , so is . ${\displaystyle P_{i}}$ ${\displaystyle \alpha =P_{i}x}$ ${\displaystyle G_{\alpha }=P_{i}^{x}}$
• The operation of on the edges is transitive. ${\displaystyle G}$
• ${\displaystyle G_{\alpha }}$operates transitive on . ${\displaystyle \Delta (\alpha )}$
• The greatest normal divisor of , which lies in, is the core of the operation of on . ${\displaystyle G}$ ${\displaystyle \bigcap P_{i}}$ ${\displaystyle G}$ ${\displaystyle \Gamma }$

sentence

The following sentence shows how the often somewhat unwieldy product property in groups can be reformulated into a simple graph-theoretical property with the help of the sub-class graph.

${\displaystyle \Gamma }$is connected if and only if is. ${\displaystyle G=\langle P_{1}\cup \cdots \cup P_{n}\rangle }$

application

The secondary class graph is used in the so-called amalgam method , in which the examination of the group is reduced to the examination of subgroups . This reduction creates advantages insofar as the group can be infinite. As long as only those are finite, all the propositions and methods of finite group theory are available. ${\displaystyle G}$ ${\displaystyle P_{i}}$ ${\displaystyle G}$ ${\displaystyle P_{i}}$

literature

• A. Delgado, D. Goldschmidt , B. Stellmacher : Groups and Graphs. New results and Methods. Birkhäuser, Basel et al. 1985, ISBN 3-7643-1736-1 ( German Mathematicians Association. DMV Seminar 6).
• Hans Kurzweil, Bernd Stellmacher: Theory of finite groups. An introduction. Springer-Verlag, Berlin et al. 1998, ISBN 3-540-60331-X ( Springer textbook ).