# Procharacteristicity is not finite-join-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., procharacteristic subgroup)notsatisfying a subgroup metaproperty (i.e., finite-join-closed subgroup property).This also implies that it doesnotsatisfy the subgroup metaproperty/metaproperties: Join-closed subgroup property (?), .

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## Contents

## 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

- Hall not implies procharacteristic
- Sylow implies procharacteristic
- 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 be .
- Let be the subgroup of defined as : the matrices where the bottom row's first two entries are zero. is a Hall subgroup of ; its order is and its index is , and is
*not*procharacteristic. - is generated by a -Sylow subgroup and a -Sylow subgroup, both of which are procharacteristic in .