Residually nilpotent group

Definition
A group is termed residually nilpotent if it satisfies the following equivalent conditions:


 * 1) Given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent
 * 2) The defining ingredient::lower central series reaches the identity element at or before the $$\omega^{th}$$ stage; in other words, the intersection of all the terms of the (finite) lower central series is the trivial group.
 * 3) The defining ingredient::nilpotent residual of the group is the trivial subgroup.

Incomparable properties

 * Hypercentral group: A residually nilpotent group need not have its upper central series go towards the group. In fact, free groups are examples of centerless residually nilpotent groups.

Metaproperties
A finite direct product of residually nilpotent groups is residually nilpotent.