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

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==Statement== | ==Statement== | ||

− | Two [[Hall subgroup]]s of the same order in a group, need not be isomorphic. | + | Two [[Hall subgroup]]s of the same order in a [[finite group]], need not be isomorphic. |

==Proof== | ==Proof== |

## 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) neednotsatisfy the second subgroup property (i.e., order-isomorphic subgroup)

View all subgroup property non-implications | View all subgroup property implications

## Statement

Two Hall subgroups of the same order in a finite 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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