Residually nilpotent group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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This is a variation of nilpotence|Find other variations of nilpotence | Read a survey article on varying nilpotence

Definition

Symbol-free definition

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

• Given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent
• The lower central series reaches the identity element in countably many steps; in other words, the intersection of all the terms of the (finite) lower central series is the trivial group

Definition with symbols

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Relation with other properties

Stronger properties

• Nilpotent group
• Free group

Weaker properties

• Hypocentral group
• Residually solvable group

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

Direct products

This group property is finite direct product-closed, viz the direct product of a finite collection of groups each having the property, also has the property
View other finite direct product-closed group properties

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