# Marginal subgroup for variety of metabelian groups

From Groupprops

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

- It is a subgroup that contains the centralizer of derived subgroup and such that the quotient group is the center of the quotient group .
- It is precisely the set of elements such that for any (possibly equal or unequal) , we have .
- It is precisely the set of elements such that for any (possibly equal or unequal) , we have .
- 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. |