# nilpotent group

We define the lower central series^{} of a group $G$ to be the filtration^{} of subgroups^{}

$$G={G}^{1}\supset {G}^{2}\supset \mathrm{\cdots}$$ |

defined inductively by:

${G}^{1}$ | $:=$ | $G,$ | ||

${G}^{i}$ | $:=$ | $[{G}^{i-1},G],i>1,$ |

where $[{G}^{i-1},G]$ denotes the subgroup of $G$ generated by all commutators^{} of the form $hk{h}^{-1}{k}^{-1}$ where $h\in {G}^{i-1}$ and $k\in G$. The group $G$ is said to be nilpotent if ${G}^{i}=1$ for some $i$.

Nilpotent groups can also be equivalently defined by means of upper central series. For a group $G$, the upper central series of $G$ is the filtration of subgroups

$${C}_{0}\subset {C}_{1}\subset {C}_{2}\subset \mathrm{\cdots}$$ |

defined by setting ${C}_{0}$ to be the trivial subgroup of $G$, and inductively taking ${C}_{i}$ to be the unique subgroup of $G$ such that ${C}_{i}/{C}_{i-1}$ is the center of $G/{C}_{i-1}$, for each $i>1$. The group $G$ is nilpotent if and only if $G={C}_{i}$ for some $i$. Moreover, if $G$ is nilpotent, then the length of the upper central series (i.e., the smallest $i$ for which $G={C}_{i}$) equals the length of the lower central series (i.e., the smallest $i$ for which ${G}^{i+1}=1$).

The *nilpotency class* or *nilpotent class* of a nilpotent group is the length of the lower central series (equivalently, the length of the upper central series).

Nilpotent groups are related to nilpotent Lie algebras in that a Lie group is nilpotent as a group if and only if its corresponding Lie algebra^{} is nilpotent. The analogy^{} extends to solvable groups^{} as well: every nilpotent group is solvable, because the upper central series is a filtration with abelian^{} quotients.

Title | nilpotent group |
---|---|

Canonical name | NilpotentGroup |

Date of creation | 2013-03-22 12:47:50 |

Last modified on | 2013-03-22 12:47:50 |

Owner | djao (24) |

Last modified by | djao (24) |

Numerical id | 8 |

Author | djao (24) |

Entry type | Definition |

Classification | msc 20F18 |

Defines | nilpotent |

Defines | upper central series |

Defines | lower central series |

Defines | nilpotency class |

Defines | nilpotent class |