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 $\mathbb {Z} /(2)\times S_{3}$ , where $S_{3}$ is the symmetric group of degree 3. There are subnormal series

\{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3},

\{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\{[0]\}\times S_{3}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3}.

$S_{3}$ contains the normal subgroup $A_{3}$ . Hence these have refinements

\{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times \{\operatorname {id} \}\;\triangleleft \;\mathbb {Z} /(2)\times A_{3}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3}

with factor groups isomorphic to $(\mathbb {Z} /(2),A_{3},\mathbb {Z} /(2))$ and

\{[0]\}\times \{\operatorname {id} \}\;\triangleleft \;\{[0]\}\times A_{3}\;\triangleleft \;\{[0]\}\times S_{3}\;\triangleleft \;\mathbb {Z} /(2)\times S_{3}

with factor groups isomorphic to $(A_{3},\mathbb {Z} /(2),\mathbb {Z} /(2))$ .

gollark: Now with optimization settings!

gollark: ```python#!/usr/bin/env python3import argparseimport subprocessparser = argparse.ArgumentParser(description='Compile a WHY program')parser.add_argument("input", help="File containing WHY source code")parser.add_argument("-o", "--output", help="Filename of the output executable to make", default="./a.why")parser.add_argument("-O", "--optimize", help="Optimization level", type=int, default="0")args = parser.parse_args()def build_C(args): template = """#define QUITELONG long long intconst QUITELONG max = @max@;int main() { QUITELONG i = 0; while (i < max) { i++; } @code@} """ for k, v in args.items(): template = template.replace(f"@{k}@", str(v)) return templateinput = args.inputoutput = args.outputtemp = "ignore-this-please"with open(input, "r") as f: contents = f.read() looplen = max(1000, (2 ** -args.optimize) * 1000000000) code = build_C({ "code": contents, "max": looplen }) with open(temp, "w") as out: out.write(code)subprocess.run(["gcc", "-x", "c", "-o", output, temp])```

gollark: ^

gollark: 937 bytes.

gollark: The WHY compiler is *very* small.

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.