Kurosch subgroup sentence

The Kurosch subgroup theorem , named after Alexander Gennadjewitsch Kurosch , is a mathematical theorem from the field of group theory . It describes the structure of subgroups of free products and represents a generalization of the Nielsen-Schreier theorem .

Formulation of the sentence

It is the free product of the groups and a subgroup. Then ${\displaystyle G={\underset {\alpha \in A}{*}}G_{\alpha }}$ ${\displaystyle G_{\alpha },\alpha \in A}$ ${\displaystyle H\leq G}$

${\displaystyle H=H_{0}*\,{\underset {\alpha \in A,d_{\alpha }\in R_{\alpha }}{*}}(H\cap (d_{\alpha }G_{\alpha }d_{\alpha }^{-1}))}$.

It is

${\displaystyle H_{0}}$ a free group,

${\displaystyle R_{\alpha }}$for each a representative system of the - double subclasses . ${\displaystyle \alpha \in A}$ ${\displaystyle (H,G_{\alpha })}$

If the index is also used , the free group has the rank ${\displaystyle [G:H]=m<\infty }$ ${\displaystyle H_{0}}$

${\displaystyle \mathrm {rg} (H_{0})=\sum _{\alpha \in A}(m-|R_{\alpha }|)+1-m}$.

Relation to the Nielsen-Schreier theorem

Kurosch's subgroup theorem is stronger than the Nielsen-Schreier theorem . The latter results from the former through specialization , as will be briefly explained here to clarify the terms. ${\displaystyle G_{\alpha }\cong \mathbb {Z} }$

If for all , the free group is of rank . A subgroup has the specified structure. With is also and therefore every trivial or likewise isomorphic to . Hence the free product of free groups and thus itself is free. So it is shown that every subgroup of a free group is free again, and that is the qualitative statement from the Nielsen-Schreier theorem. ${\displaystyle G_{\alpha }\cong \mathbb {Z} }$ ${\displaystyle \alpha \in A}$ ${\displaystyle G={\underset {\alpha \in A}{*}}G_{\alpha }}$ ${\displaystyle |A|}$ ${\displaystyle H}$ ${\displaystyle G_{\alpha }\cong \mathbb {Z} }$ ${\displaystyle d_{\alpha }G_{\alpha }d_{\alpha }^{-1}\cong \mathbb {Z} }$ ${\displaystyle H\cap (d_{\alpha }G_{\alpha }d_{\alpha }^{-1})}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle H}$

For the quantitative statement of the Nielsen-Schreier theorem, we restrict ourselves to a finite index set . Let the infinite cyclic group be generated by. Since the index of in is finite, the minor classes cannot all be different. There must therefore be a with and therefore also a with There , is , so ${\displaystyle A}$ ${\displaystyle G_{\alpha }}$ ${\displaystyle g_{\alpha }\in G_{\alpha }}$ ${\displaystyle H}$ ${\displaystyle G}$ ${\displaystyle Hd_{\alpha }g_{\alpha }^{r},\,r>0}$ ${\displaystyle r>0}$ ${\displaystyle Hd_{\alpha }=Hd_{\alpha }g_{\alpha }^{r}}$ ${\displaystyle h\in H}$ ${\displaystyle hd_{\alpha }=d_{\alpha }g_{\alpha }^{r}}$ ${\displaystyle r>0}$ ${\displaystyle h\not =1}$

${\displaystyle 1\not =h=d_{\alpha }g_{\alpha }^{r}d_{\alpha }^{-1}\in H\cap (d_{\alpha }G_{\alpha }d_{\alpha }^{-1})}$

So this subgroup is not trivial and therefore isomorphic to . So is ${\displaystyle \mathbb {Z} }$

${\displaystyle \mathrm {rg} (H)=\mathrm {rg} (H_{0})+\sum _{\alpha \in A,d_{\alpha }\in R_{\alpha }}\underbrace {\mathrm {rg} (H\cap (d_{\alpha }G_{\alpha }d_{\alpha }^{-1}))} _{=1}}$     (since the ranks of free groups add up for free products)

${\displaystyle =\quad \sum _{\alpha \in A}(m-|R_{\alpha }|)+1-m\,\,+\,\,\sum _{\alpha \in A}|R_{\alpha }|}$     (according to the ranking formula from Kurosch's subgroup theorem)

${\displaystyle =\quad |A|m+1-m}$,

and that is exactly the formula from Nielsen-Schreier's theorem.

Individual evidence

1. ^ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , sentence 6.3.1 .: The Kuroš Subgroup Theorem
2. ↑ Wilfried Imrich in Combinatorial Mathematics V , Springer Verlag (1976), Lecture Notes in Mathematics 622, Subgroups and Graphs , Chapter 9: The Kurosh Subgroup Theorem
3. ^ Wilhelm Specht : Group theory. Springer-Verlag (1956), chapter 2.2.2, sentence 8
4. ↑ DJS Robinson : A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 978-1-4612-6443-9 , explanations for theorem 6.3.1.