Hall not implies order-isomorphic

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property need not satisfy the second subgroup property
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Property "Page" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
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  • Property "Satisfies property" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.

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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