PlanetMath: nilpotent group

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nilpotent group

(Definition)

We define the lower central series of a group to be the filtration of subgroups

$\displaystyle G = G^1 \supset G^2 \supset \cdots$

defined inductively by:

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

where $[G^{i-1},G]$ denotes the subgroup of

generated by all commutators of the form $hkh^{-1}k^{-1}$ where $h \in G^{i-1}$ and $k \in G$ . The group

is said to be nilpotent if

for some

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

$\displaystyle C_0 \subset C_1 \subset C_2 \subset \cdots$

defined by setting

to be the trivial subgroup of

, and inductively taking

to be the unique subgroup of

such that $C_i/C_{i-1}$ is the center of $G/C_{i-1}$ , for each

. The group

is nilpotent if and only if

for some

. Moreover, if

is nilpotent, then the length of the upper central series (i.e., the smallest

for which

) equals the length of the lower central series (i.e., the smallest

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.

"nilpotent group" is owned by djao.

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Also defines:

nilpotent, upper central series, lower central series, nilpotency class, nilpotent class

Attachments:: characterization of finite nilpotent groups (Theorem) by yark
finite nilpotent groups (Topic) by Algeboy

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Cross-references: quotients, abelian, solvable groups, Lie algebra, Lie group, nilpotent Lie algebras, length, center, trivial subgroup, commutators, generated by, subgroups, filtration, group
There are 22 references to this entry.

This is version 5 of nilpotent group, born on 2002-06-16, modified 2006-11-01.
Object id is 3113, canonical name is NilpotentGroup.
Accessed 12779 times total.

Classification:

AMS MSC:

20F18 (Group theory and generalizations :: Special aspects of infinite or finite groups :: Nilpotent groups)

Pending Errata and Addenda

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