Poincaré's theorem (group theory)
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 .
- Then the following statements apply:
- (i)
- (ii) If the in finite all index, has its average self finite index.
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 .
- 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 .
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 ).