# Direct factor is not finite-join-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor)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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## Statement

A join of finitely many direct factors of a group need not be a direct factor. More specifically, it is possible to have a group and two subgroups of such that both and are direct factors and the join is not a direct factor.

## Related facts

## Proof

### An abelian group example

Suppose denotes the cyclic group of order . Define:

.

Consider the following subgroups:

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Then, both and are direct factors of , with a common direct factor complement . On the other hand, we have:

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This is not a direct factor of , because if a complement exists, it must have order two, but all elements of outside have order four.