Let be a property of groups, assumed to be an isomorphic invariant (that is, if a group has property , then every group isomorphic to also has property ). We shall sometimes refer to groups with property as -groups.
A group is said to be residually if for every there is a normal subgroup of such that and has property . Equivalently, is residually if and only if
where means that is normal in and has property .
It can be shown that a group is residually if and only if it is isomorphic to a subdirect product of -groups. If is a hereditary property (that is, every subgroup (http://planetmath.org/Subgroup) of an -group is an -group), then a group is residually if and only if it can be embedded in an unrestricted direct product of -groups.
It can be shown that a group is residually solvable if and only if the intersection of the derived series of is trivial (see transfinite derived series). Similarly, a group is residually nilpotent if and only if the intersection of the lower central series of is trivial.
|Date of creation||2013-03-22 14:53:22|
|Last modified on||2013-03-22 14:53:22|
|Last modified by||yark (2760)|