Hall not implies procharacteristic

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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., automorph-conjugate subgroup)
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Statement

A Hall subgroup of a group need not be procharacteristic.

Definitions used

Hall subgroup

Further information: Hall subgroup

A subgroup of a finite group is termed a Hall subgroup if its order and index are relatively prime.

Procharacteristic subgroup

Further information: Procharacteristic subgroup

A subgroup H of a finite group G is termed a procharacteristic subgroup if for any automorphism \sigma of G, H and \sigma(H) are conjugate in the subgroup \langle H, \sigma(H)\rangle.

Related facts

Similar facts

Applications

Results of the opposite kind

Proof

General setup

We prove that if r is an odd prime, q is a power of a prime p, and gcd(r,q-1) = 1, then any subgroup of index (q^r - 1)/(q-1) in SL(r,q) is a Hall subgroup.

This follows from order computation.

Now observe that the parabolic subgroup P_{r-1,1} has the required index, and hence is a Hall subgroup. By P_{r-1,1} we mean the subgroup of SL(r,q) comprising those elements where the bottom row has only one nonzero entry, namely the last.

Now consider P_{r-1,1} and its image under the transpose-inverse automorphism \tau. For r > 2 (which is true if r is an odd prime, the transpose-inverse has an invariant one-dimensional subspace while the original subgroup doesn't. Hence, the two subgroups cannot be conjugate.

Now, it is further true that both subgroups are maximal in the whole group. Thus, the subgroup generated by P_{r-1,1} and its image under \tau is the whole group. In particular, we set:

  • The whole group is SL(r,q).
  • The Hall subgroup is P_{r-1,1}.
  • The automorphism is \tau, the transpose-inverse automorphism.
  • The Hall subgroup P_{r-1,1} and its image under the automorphism \tau generate the whole group, but they are not conjugate in the whole group. Thus, the condition for procharacteristicity is violated.

A specific example

A specific example is where the group is SL(3,2), which has order 168. In this case, the Hall subgroup is isomorphic to the symmetric group on four elements, which has order 24 and index 7. Its image under the transpose-inverse automorphism is another subgroup of order 24, and they together generate the whole group. However, they are not conjugate in the whole group.