# Difference between revisions of "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, 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 $SL(2,11)$. 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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