Finite direct product-closed group property

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This article defines a group metaproperty: a property that can be evaluated to true/false for any group property
View a complete list of group metaproperties

Definition

A group property \alpha is termed finite direct product-closed if it satisfies the following equivalent conditions:

  1. Whenever G_1 and G_2 are groups satisfying \alpha, the external direct product G_1 \times G_2 also satisfies \alpha.
  2. For any positive integer \alpha and groups G_1,G_2,\dots,G_n all of which satisfy \alpha, the external direct product G_1 \times G_2 \times \dots \times G_n also satisfies \alpha.

Note that if the trivial group also satisfies \alpha, we say that \alpha is strongly finite direct product-closed.

Relation with other metaproperties

Stronger metaproperties

Metaproperty Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
direct product-closed group property |FULL LIST, MORE INFO
varietal group property Pseudovarietal group property|FULL LIST, MORE INFO
quasivarietal group property |FULL LIST, MORE INFO
pseudovarietal group property |FULL LIST, MORE INFO