# Marginal subgroup for variety of metabelian groups

This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions

## Definition

Let $G$ be a group. The marginal subgroup for variety of metabelian groups for $G$ is defined in the following equivalent ways:

1. It is a subgroup $H$ that contains the centralizer of derived subgroup $C_G([G,G])$ and such that the quotient group $H/C_G([G,G])$ is the center of the quotient group $G/C_G([G,G])$.
2. It is precisely the set of elements $g \in G$ such that for any (possibly equal or unequal) $x_1,x_2,x_3,x_4 \in G$, we have $[[x_1,x_2],[x_3,x_4]] = [[gx_1,x_2],[x_3,x_4]]$.
3. It is precisely the set of elements $g \in G$ such that for any (possibly equal or unequal) $x_1,x_2,x_3,x_4 \in G$, we have $[[x_1,x_2],[x_3,x_4]] = [[gx_1,x_2],[x_3,x_4]] = [[x_1,gx_2],[x_3,x_4]] =[[x_1,x_2],[gx_3,x_4]] =[[x_1,x_2],[x_3,gx_4]]$.
4. It is the marginal subgroup corresponding to the variety of metabelian groups.

## Relation with other subgroup-defining functions

### Smaller subgroup-defining functions

Subgroup-defining function Meaning
center commutes with all elements
second center second member of upper central series, also, marginal subgroup for nilpotency class two
centralizer of derived subgroup centralizer of derived subgroup
third center third member of upper central series, also, marginal subgroup for nilpotency class three.