Variety-containing subgroup

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed a variety-containing subgroup if there exists a subvariety $\mathcal{V}$ of the variety of groups such that:

• $H\in mathcalV$.
• If $K\le G$ is such that $K\in \mathcal{V}$, then $K\le H$.

Relation with other properties

Weaker properties

• Homomorph-containing subgroup
• Intermediately fully characteristic subgroup
• Fully characteristic subgroup
• Strictly characteristic subgroup
• Intermediately characteristic subgroup
• Characteristic subgroup
• Homomorph-dominating subgroup
• Isomorph-containing subgroup
• Isomorph-dominating subgroup