Procharacteristicity is not finite-join-closed

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) uses::Hall not implies procharacteristic
 * 2) uses::Sylow implies procharacteristic
 * 3) uses::Hall implies join of Sylow subgroups

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 $$G$$.