Hall not implies order-isomorphic
From Groupprops
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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EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property {{{stronger}}} but not {{{weaker}}}|View examples of subgroups satisfying property {{{stronger}}} and {{{weaker}}}
- Property "Satisfies property" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
- Property "Dissatisfies 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.
- Property "Satisfies property" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
- 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 
. This group has order 
, and the 
-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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