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

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{{subgroup property non-implication}}
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{{subgroup property non-implication in|
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group property = finite group|
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stronger = Hall subgroup|
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weaker = order-isomorphic subgroup}}
  
 
==Statement==
 
==Statement==

Revision as of 10:58, 8 August 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)
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Statement

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

Proof

An example is the group SL(2,11). This group has order 2^2.3.11, and the 11'-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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