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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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions
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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In terms of property operators
In terms of the residually operator
This property is obtained by applying the residually operator to the property: nilpotent group
View other properties obtained by applying the residually operator
Relation with other properties
Stronger properties
Weaker properties
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.