residually 𝔛
Let 𝔛 be a property of groups, assumed to be an isomorphic 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 x∈G\{1} there is a normal subgroup N of G such that x∉N and G/N has property 𝔛. Equivalently, G is residually 𝔛 if and only if
⋂N⊴𝔛GN={1}, |
where N⊴𝔛G 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 subgroup (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 intersection 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 |