Procharacteristicity is not finite-join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., procharacteristic subgroup) not satisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .
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Statement

Verbal statement

A join of two procharacteristic subgroups of a group need not be procharacteristic. Equivalently, a join of finitely many procharacteristic subgroups need not be procharacteristic.

Related facts

• Automorph-conjugacy is not finite-join-closed
• Intermediate automorph-conjugacy is not finite-join-closed

Facts used

1. Hall not implies procharacteristic
2. Sylow implies procharacteristic
3. Hall implies join of Sylow subgroups

Proof

Property-theoretic proof

The proof directly follows by combining facts (1), (2) and (3).

Hands-on proof

• Let $G$ be $SL(3,2)$.
• Let $L$ be the subgroup of $G$ defined as $P_{r-1,1}$: the matrices where the bottom row's first two entries are zero. $L$ is a Hall subgroup of $G$; its order is $24$ and its index is $7$, and is not procharacteristic.
• $L$ is generated by a $2$-Sylow subgroup and a $3$-Sylow subgroup, both of which are procharacteristic in $G$.