Difference between revisions of "Composition factor-equivalent groups"

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==Definition==
 
==Definition==
  
Suppose <math>G</math> and <math>H</math> are [[defining ingredient::group of finite composition length|groups of finite composition length]]. We say that <math>G</math> and <math>H</math> are '''composition factor-equivalent''' if the lists of composition factors of <math>G</math> and <math>H</math> are the same (possibly, occurring in a different order). In other words, every [[defining ingredient::simple group]] has the same number of isomorphic copies in the [[defining ingredient::composition series]] of <math>G</math> as in the composition series of <math>H</math>.
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Suppose <math>G</math> and <math>H</math> are [[defining ingredient::group of finite composition length|groups of finite composition length]]. We say that <math>G</math> and <math>H</math> are '''composition factor-equivalent''' if the multisets of composition factors of <math>G</math> and <math>H</math> are the same (possibly, occurring in a different order). In other words, every [[defining ingredient::simple group]] has the same number of isomorphic copies in the [[defining ingredient::composition series]] of <math>G</math> as in the composition series of <math>H</math>.
  
 
==Facts==
 
==Facts==

Latest revision as of 21:01, 7 June 2012

This article defines an equivalence relation over the collection of groups. View a complete list of equivalence relations on groups.

Definition

Suppose G and H are groups of finite composition length. We say that G and H are composition factor-equivalent if the multisets of composition factors of G and H are the same (possibly, occurring in a different order). In other words, every simple group has the same number of isomorphic copies in the composition series of G as in the composition series of H.

Facts

  • The trivial group is not composition factor-equivalent to any other group.
  • A simple group is not composition factor-equivalent to any other group.
  • Any two finite groups that are composition factor-equivalent must have the same order.
  • Two finite solvable groups are composition factor-equivalent if and only if they have the same order.