# Contranormality is upper join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., contranormal subgroup) satisfying a subgroup metaproperty (i.e., upper join-closed subgroup property)

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## Contents

## Statement

### Statement with symbols

Suppose is a subgroup, and , is an indexed family of subgroups with for each . Then, if is contranormal in each , is also contranormal in the [join of subgroups|join]] of the s.

## Definitions used

### Contranormal subgroup

`Further information: contranormal subgroup`

is a contranormal subgroup if for any containing such that is normal in , .

## Related facts

## Facts used

- Normality satisfies transfer condition: If is a normal subgroup, and , then is normal in .

## Proof

**Given**: , family of subgroups with , and contranormal in each .

**To prove**: is normal in the join of all the s.

**Proof**: Suppose is a normal subgroup of the join of the s, containing . Then, for each , is a subgroup of containing , and by fact (1), it is normal in . Since is contranormal in , for each , so for each . Thus, must equal the join of the s.