Hall not implies procharacteristic

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 $\langle H, \sigma(H)\rangle$.

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.