# Difference between revisions of "Nilpotent residual"

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The nilpotent residual of a group <math>G</math> is denoted <math>\gamma_\omega(G)</math> or <math>\gamma_\infty(G)</math>. | The nilpotent residual of a group <math>G</math> is denoted <math>\gamma_\omega(G)</math> or <math>\gamma_\infty(G)</math>. | ||

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+ | ==Related notions== | ||

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+ | * [[Hypocenter]] is the ultimate stable member of the transfinite [[lower central series]]. For a [[finite group]] or a [[virtually nilpotent group]], the hypocenter coincides with the nilpotent residual. | ||

+ | * [[Lower Fitting series]] for a [[finite group]] is a Fitting series where each successive member is the nilpotent residual of its predecessor. |

## Latest revision as of 14:45, 2 August 2011

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

The **nilpotent residual** of a group can be defined in the following equivalent ways:

- It is the intersection of all nilpotent-quotient normal subgroups of the group (i.e., normal subgroups such that the quotient is a nilpotent group).
- It is the intersection of all members of the (finite) lower central series of the group. In particular, it is the member of the transfinite lower central series.

A group is residually nilpotent if and only if its nilpotent residual is trivial.

When the group is a finite group or a virtually nilpotent group, and in some other nice situations, the nilpotent residual of a group itself is a nilpotent-quotient normal subgroup. If this is the case, it can also be described as the *unique* smallest nilpotent-quotient normal subgroup.

The nilpotent residual of a group is denoted or .

## Related notions

- Hypocenter is the ultimate stable member of the transfinite lower central series. For a finite group or a virtually nilpotent group, the hypocenter coincides with the nilpotent residual.
- Lower Fitting series for a finite group is a Fitting series where each successive member is the nilpotent residual of its predecessor.