# residually $\U0001d51b$

Let $\U0001d51b$ be a property of groups,
assumed to be an isomorphic^{} invariant
(that is, if a group $G$ has property $\U0001d51b$,
then every group isomorphic to $G$ also has property $\U0001d51b$).
We shall sometimes refer to groups with property $\U0001d51b$ as $\U0001d51b$-groups.

A group $G$ is said to be *residually $\mathrm{X}$*
if for every $x\in G\backslash \{1\}$ there is a normal subgroup^{} $N$ of $G$
such that $x\notin N$ and $G/N$ has property $\U0001d51b$.
Equivalently, $G$ is residually $\U0001d51b$ if and only if

$$\bigcap _{N{\mathrm{\u22b4}}_{\U0001d51b}G}N=\{1\},$$ |

where $N{\mathrm{\u22b4}}_{\U0001d51b}G$ means that $N$ is normal in $G$ and $G/N$ has property $\U0001d51b$.

It can be shown that a group is residually $\U0001d51b$
if and only if it is isomorphic to a subdirect product of $\U0001d51b$-groups.
If $\U0001d51b$ is a hereditary property
(that is, every subgroup^{} (http://planetmath.org/Subgroup) of an $\U0001d51b$-group is an $\U0001d51b$-group),
then a group is residually $\U0001d51b$ if and only if
it can be embedded in an unrestricted direct product of $\U0001d51b$-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 $\U0001d51b$ |
---|---|

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 |