PlanetMath: residually $\mathfrak{X}$

(more info)

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residually $\mathfrak{X}$

(Definition)

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

A group is said to be residually $\mathfrak{X}$ if for every $x\in G\backslash\{1\}$ there is a normal subgroup of such that $x\notin N$ and has property $\mathfrak{X}$ . Equivalently, is residually $\mathfrak{X}$ if and only if

$\begin{displaymath} \bigcap_{N\trianglelefteq _\mathfrak{X}G}\!\!N=\{1\}, \end{displaymath}$

where $N\trianglelefteq _\mathfrak{X}G$ means that

is normal in

and

has property $\mathfrak{X}$ .

It can be shown that a group is residually $\mathfrak{X}$ if and only if it is isomorphic to a subdirect product of $\mathfrak{X}$ -groups. If $\mathfrak{X}$ is a hereditary property (that is, every subgroup of an $\mathfrak{X}$ -group is an $\mathfrak{X}$ -group), then a group is residually $\mathfrak{X}$ if and only if it can be embedded in an unrestricted direct product of $\mathfrak{X}$ -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.