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 Z/(2)×S3displaystyle mathbb Z /(2)times S_3mathbb Z/(2)times S_3, where S3displaystyle S_3S_3 is the symmetric group of degree 3. There are subnormal series


[0]×id◃Z/(2)×id◃Z/(2)×S3,displaystyle [0]times operatorname id ;triangleleft ;mathbb Z /(2)times operatorname id ;triangleleft ;mathbb Z /(2)times S_3,[0]times operatorname id;triangleleft ;mathbb Z/(2)times operatorname id;triangleleft ;mathbb Z/(2)times S_3,

[0]×id◃[0]×S3◃Z/(2)×S3.displaystyle [0]times operatorname id ;triangleleft ;[0]times S_3;triangleleft ;mathbb Z /(2)times S_3.[0]times operatorname id;triangleleft ;[0]times S_3;triangleleft ;mathbb Z/(2)times S_3.

S3displaystyle S_3S_3 contains the normal subgroup A3displaystyle A_3A_3. Hence these have refinements


[0]×id◃Z/(2)×id◃Z/(2)×A3◃Z/(2)×S3displaystyle [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[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 (Z/(2),A3,Z/(2))displaystyle (mathbb Z /(2),A_3,mathbb Z /(2))(mathbb Z/(2),A_3,mathbb Z/(2)) and


[0]×id◃[0]×A3◃[0]×S3◃Z/(2)×S3displaystyle [0]times operatorname id ;triangleleft ;[0]times A_3;triangleleft ;[0]times S_3;triangleleft ;mathbb Z /(2)times S_3[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 (A3,Z/(2),Z/(2))displaystyle (A_3,mathbb Z /(2),mathbb Z /(2))(A_3,mathbb Z/(2),mathbb Z/(2)).



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.mw-parser-output cite.citationfont-style:inherit.mw-parser-output qquotes:"""""""'""'".mw-parser-output code.cs1-codecolor:inherit;background:inherit;border:inherit;padding:inherit.mw-parser-output .cs1-lock-free abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-lock-subscription abackground:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registrationcolor:#555.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration spanborder-bottom:1px dotted;cursor:help.mw-parser-output .cs1-hidden-errordisplay:none;font-size:100%.mw-parser-output .cs1-visible-errorfont-size:100%.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-formatfont-size:95%.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-leftpadding-left:0.2em.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-rightpadding-right:0.2em

  • Rotman, Joseph (1994). An introduction to the theory of groups. New York: Springer-Verlag. ISBN 0-387-94285-8.



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