Pseudovarietal 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 pseudovarietal if it satisfies the following three conditions:

  1. It is a subgroup-closed group property, i.e., whenever G is a group satisfying \alpha and H is a subgroup of G, H also satisfies \alpha.
  2. It is a quotient-closed group property, i.e., whenever G is a group satisfying \alpha and H is a normal subgroup of G, the quotient group G/H also satisfies \alpha.
  3. It is a finite direct product-closed group property, i.e., whenever G_1,G_2,\dots,G_n are groups all of which satisfy \alpha, the external direct product G_1 \times G_2 \times \dots \times G_n also satisfies \alpha.

Relation with other metaproperties

Stronger metaproperties

Metaproperty Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
varietal group property subgroup-closed, quotient-closed, and closed under arbitrary direct products |FULL LIST, MORE INFO

Weaker metaproperties

Metaproperty Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
subgroup-closed group property closed under taking subgroups |FULL LIST, MORE INFO
quotient-closed group property closed under taking quotient groups |FULL LIST, MORE INFO
finite direct product-closed group property closed under taking finite direct products |FULL LIST, MORE INFO