Direct product-closed subgroup property

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This article defines a subgroup metaproperty: a property that can be evaluated to true/false for any subgroup property
View a complete list of subgroup metaproperties
View subgroup properties satisfying this metaproperty| View subgroup properties dissatisfying this metaproperty
VIEW RELATED: subgroup metaproperty satisfactions| subgroup metaproperty dissatisfactions


BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
This article is about a general term. A list of important particular cases (instances) is available at Category:Direct product-closed subgroup properties

Definition

Definition with symbols

A subgroup property p is termed direct product-closed if, given a nonempty indexing set I, and a collection of group-subgroup pairs H_i \le G_i for i \in I, such that each H_i satisfies p as a subgroup of G_i, then the direct product of the H_is satisfies p via its natural embedding in the direct product of G_is.