residually 𝔛


Let 𝔛 be a property of groups, assumed to be an isomorphicPlanetmathPlanetmathPlanetmath invariant (that is, if a group G has property 𝔛, then every group isomorphic to G also has property 𝔛). We shall sometimes refer to groups with property 𝔛 as 𝔛-groups.

A group G is said to be residually X if for every xG\{1} there is a normal subgroupMathworldPlanetmath N of G such that xN and G/N has property 𝔛. Equivalently, G is residually 𝔛 if and only if

N𝔛GN={1},

where N𝔛G means that N is normal in G and G/N has property 𝔛.

It can be shown that a group is residually 𝔛 if and only if it is isomorphic to a subdirect product of 𝔛-groups. If 𝔛 is a hereditary property (that is, every subgroupMathworldPlanetmathPlanetmath (http://planetmath.org/Subgroup) of an 𝔛-group is an 𝔛-group), then a group is residually 𝔛 if and only if it can be embedded in an unrestricted direct product of 𝔛-groups.

It can be shown that a group G is residually solvable if and only if the intersectionMathworldPlanetmathPlanetmath of the derived series of G is trivial (see transfinite derived series). Similarly, a group G is residually nilpotent if and only if the intersection of the lower central series of G is trivial.

Title residually 𝔛
Canonical name ResiduallymathfrakX
Date of creation 2013-03-22 14:53:22
Last modified on 2013-03-22 14:53:22
Owner yark (2760)
Last modified by yark (2760)
Numerical id 15
Author yark (2760)
Entry type Definition
Classification msc 20E26
Related topic AGroupsEmbedsIntoItsProfiniteCompletionIfAndOnlyIfItIsResiduallyFinite
Defines residually finite
Defines residually nilpotent
Defines residually solvable
Defines residually soluble