Nilpotent residual: Difference between revisions
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The '''nilpotent residual''' of a group can be defined in the following equivalent ways: | The '''nilpotent residual''' of a group can be defined in the following equivalent ways: | ||
# It is the intersection of all [[defining ingredient::nilpotent-quotient normal subgroup]]s of the group (i.e., [[normal subgroup]]s such that the quotient is a [[nilpotent group]]). | # It is the intersection of all [[defining ingredient::nilpotent-quotient normal subgroup]]s of the group (i.e., [[defining ingredient::normal subgroup]]s such that the quotient is a [[defining ingredient::nilpotent group]]). | ||
# It is the intersection of all members of the (finite) [[defining ingredient::lower central series]] of the group. In particular, it is the <math>\omega^{th}</math> member of the transfinite lower central series. | # It is the intersection of all members of the (finite) [[defining ingredient::lower central series]] of the group. In particular, it is the <math>\omega^{th}</math> member of the transfinite lower central series. | ||
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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. | 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 <math>G</math> is denoted <math>\gamma_\omega(G)</math> or <math>\gamma_\infty(G)</math>. | |||
==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. | |||
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.