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.
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* [[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
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Definition

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

  1. 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).
  2. It is the intersection of all members of the (finite) lower central series of the group. In particular, it is the \omega^{th} 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 G is denoted \gamma_\omega(G) or \gamma_\infty(G).

Related notions