Schreier refinement theorem

In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an isomorphic one.

The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. Baumslag (2006) gives a short proof by intersecting the terms in one subnormal series with those in the other series.

Example

Consider , where is the symmetric group of degree 3. There are subnormal series

contains the normal subgroup . Hence these have refinements

with factor groups isomorphic to and

with factor groups isomorphic to .

gollark: From what? They have no context except a bunch of code they also can't read.
gollark: Most useful access to it requires an account. Nobody knows how to make one, especially as the authentication mechanisms it relied on are all down, but fortunately a "try APL" REPL with more permissions than it probably should have still functions and allows anonymous access.
gollark: Well, in my headcanon, the system was never designed to be "magic" but is a relic from a more advanced civilisation which can self-repair a decent amount.
gollark: Oh wait, you can, have the system also have a bunch of robotic lifeforms tied into it but make them weird lifeishly and call them "elementals".
gollark: I don't think you can give this system many powers unless you just handwave it as magic nanobots or something.

References

  • Baumslag, Benjamin (2006), "A simple way of proving the Jordan-Hölder-Schreier theorem", American Mathematical Monthly, 113 (10): 933–935, doi:10.2307/27642092
  • Rotman, Joseph (1994). An introduction to the theory of groups. New York: Springer-Verlag. ISBN 0-387-94285-8.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.