Quasiverbal subgroup

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This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

Definition

Suppose \mathcal{V} is a subquasivariety of the variety of groups. Equivalently, the group property of being in \mathcal{V} is a quasivarietal group property.

The \mathcal{V}-quasiverbal subgroup of a group G is defined in the following equivalent ways:

Metaproperties

Metaproperty name Satisfied? Proof Statement with symbols
quotient-transitive subgroup property Yes quasiverbality is quotient-transitive Suppose H \le K \le G are groups such that H is a quasiverbal subgroup of G and K/H is a quasiverbal subgroup of G/H. Then, K is a quasiverbal subgroup of G.
strongly join-closed subgroup property Yes quasiverbality is strongly join-closed Suppose H_i, i \in I are all quasiverbal subgroups of a group G. Then, the join of subgroups \langle H_i \rangle_{i \in I} is also a quasiverbal subgroup.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
verbal subgroup similar definition, but for a subvariety instead of a subquasivariety verbal implies quasiverbal quasiverbal not implies verbal |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
quotient-subisomorph-containing subgroup contained in the kernel of any homomorphism to the quotient group |FULL LIST, MORE INFO
fully invariant subgroup invariant under all endomorphisms [(via quotient-subisomorph-containing) (via quotient-subisomorph-containing) Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO
characteristic subgroup invariant under all automorphisms (via fully invariant) (via fully invariant) Quotient-subisomorph-containing subgroup|FULL LIST, MORE INFO