# Hirsch-Plotkin radical

From Groupprops

## Definition

The **Hirsch-Plotkin radical** of a group can be defined in the following equivalent ways:

- It is the subgroup generated by all the locally nilpotent normal subgroups (i.e., normal locally nilpotent subgroups) of the group.
- It is the unique largest locally nilpotent normal subgroup of the group.
- It is the subgroup generated by all the locally nilpotent characteristic subgroups (i.e., characteristic locally nilpotent subgroups) of the group.
- It is the unique largest locally nilpotent characteristic subgroup of the group.

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

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Hirsch-Plotkin radical, all facts related to Hirsch-Plotkin radical) |Survey articles about this | Survey articles about definitions built on this

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View a list of other standard non-basic definitions

## Relation with other subgroup-defining functions

### Smaller subgroup-defining functions

- Fitting subgroup (they become equal in case the group is finite)