Direct factor is not finite-join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., direct factor) 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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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.
An abelian group example
Suppose denotes the cyclic group of order . Define:
Consider the following subgroups:
Then, both and are direct factors of , with a common direct factor complement . On the other hand, we have:
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.