Epimarginal subgroup

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Definition

Suppose ${\mathcal {V}}$ is a subvariety of the variety of groups. Consider a group $G$ (not necessarily in ${\mathcal {V}}$. The epimarginal subgroup of $G$ with respect to ${\mathcal {V}}$ is defined as follows:

• Consider group extensions $1\to A\to E\to G\to 1$ with the property that the image of $A$ in $E$ in contained in the ${\mathcal {V}}$-marginal subgroup of $E$.
• For each such extension, consider the image in $G$ under the map $E\to G$ of the ${\mathcal {V}}$-marginal subgroup of $E$.
• Take the intersection of all possible such images over all possible such extensions.

The intersection thus obtained is the ${\mathcal {V}}$-epimarginal subgroup of $G$. Any subgroup that arises as the ${\mathcal {V}}$-epimarginal subgroup for some variety ${\mathcal {V}}$ is termed an epimarginal subgroup.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup		epimarginal implies characteristic	characteristic not implies epimarginal	|FULL LIST, MORE INFO
normal subgroup		(via characteristic)	(via characteristic)	|FULL LIST, MORE INFO