Poincaré's theorem (group theory)

The title of this article is ambiguous. The present sentence is only one of several sentences associated with Poincaré's name. See Poincaré's Theorem (BKL)!

Among the many results that Henri Poincaré in different areas of mathematics has contributed belongs in group theory one as a set of Poincaré designated theorem , in the Poincaré one fundamental question to indexes of subgroups treated.

formulation

The sentence can be summarized as follows:

Let a group be given and within it a finite number of subgroups . ${\displaystyle G}$ ${\displaystyle U_{1},\ldots ,U_{n}\leq G\;(n\in \mathbb {N} )}$

Then the following statements apply:

(i) ${\displaystyle (G\colon \bigcap _{i=1}^{n}{U_{i}})\leq \prod _{i=1}^{n}(G\colon U_{i})}$

(ii) If the in finite all index, has its average self finite index. ${\displaystyle U_{i}\;(i=1,\ldots ,n)}$ ${\displaystyle G}$ ${\displaystyle \bigcap _{i=1}^{n}{U_{i}}}$

Remarks

• The basic estimation in (i) results directly from the fact that for two sub-groups , and each - coset the equation satisfied. In this way one immediately gains the aforementioned estimate for the case , which can then be extended to the general case by complete induction . ${\displaystyle U_{1},U_{2}\leq G}$ ${\displaystyle x\in G}$ ${\displaystyle U_{1}\cap U_{2}}$ ${\displaystyle \left(U_{1}\cap U_{2}\right)x={U_{1}}x\cap {U_{2}}x}$ ${\displaystyle n=2}$
• Under certain conditions, the equals sign applies to (i) above . If there are about two subgroups whose indices are finite in both and at the same time coprime , then it even applies . ${\displaystyle U_{1},U_{2}\leq G}$ ${\displaystyle G}$ ${\displaystyle (G\colon U_{1}\cap U_{2})=(G\colon U_{1})\cdot (G\colon U_{2})}$

literature

• AG Kurosch : Group Theory I . Edited in German by Dr. Reinhard Strecker (=  Mathematical Textbooks and Monographs, Department I, Mathematical Textbooks . Volume III / I ). 2nd, revised and expanded edition. Akademie-Verlag , Berlin 1970 ( MR0266978 ).
• Kurt Meyberg : Algebra. Part 1 (=  Mathematical basics for mathematicians, physicists and engineers. ). Carl Hanser Verlag , Munich, Vienna 1975, ISBN 3-446-11965-5 ( MR0460010 ).
• Hans Schwerdtfeger : Introduction to Group Theory . Noordhoff International Publishing, Leyden 1976, ISBN 90-286-0495-2 ( MR0435190 ).

Individual evidence

1. ↑ ^{a b} A. G. Kurosch: Group theory I. 1970, p. 42
2. ↑ ^{a b c d} Kurt Meyberg: Algebra. Part 1. 1975, p. 50
3. ^ ^{A b} Hans Schwerdtfeger: Introduction to Group Theory. 1976, p. 64