Nilpotent residual

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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 ${\gamma }_{\mathrm{\infty }}(G)$.

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.