Procharacteristicity is not finite-join-closed

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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

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.