Difference between revisions of "Hall not implies order-isomorphic"

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==Proof==
 
==Proof==
  
An example is the group <math>SL(2,11)</math>. This group has order <math>2^2.3.5.11</math>, and the <math>\{ 2,3 \}</math>-Hall subgroups have order 12. It turns out that there are two possible isomorphism classes of such subgroups: the [[dihedral group]] on twelve elements, and [[alternating group:A4|the alternating group on four elements]].
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An example is the group <math>PSL(2,11)</math>. This group has order <math>2^2.3.5.11</math>, and the <math>\{ 2,3 \}</math>-Hall subgroups have order 12. It turns out that there are two possible isomorphism classes of such subgroups: the [[dihedral group]] on twelve elements, and [[alternating group:A4|the alternating group on four elements]].
  
 
Here are explicit embeddings of these subgroups:
 
Here are explicit embeddings of these subgroups:
  
 
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{{fillin}}

Revision as of 19:52, 22 November 2008

This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a finite group, every subgroup satisfying the first subgroup property (i.e., Hall subgroup) need not satisfy the second subgroup property (i.e., order-isomorphic subgroup)
View all subgroup property non-implications | View all subgroup property implications

Statement

Two Hall subgroups of the same order in a finite group, need not be isomorphic.

Proof

An example is the group PSL(2,11). This group has order 2^2.3.5.11, and the \{ 2,3 \}-Hall subgroups have order 12. It turns out that there are two possible isomorphism classes of such subgroups: the dihedral group on twelve elements, and the alternating group on four elements.

Here are explicit embeddings of these subgroups:

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