Hall not implies order-isomorphic: Difference between revisions

Revision as of 19:45, 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)
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Statement

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

Proof

An example is the group $S L (2, 11)$ . 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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 ==Proof==
-An example is the group <math>SL(2,11)</math>. This group has order <math>2^2.3.11</math>, and the <math>11'</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]].
+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]].
 Here are explicit embeddings of these subgroups:
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