Nilpotency class: Difference between revisions

Latest revision as of 17:14, 17 August 2013

This article defines an arithmetic function on a restricted class of groups, namely: nilpotent groups

Definition

Equivalent definitions in tabular format

The notion of nilpotency class (also called nilpotence class) makes sense for any nilpotent group and is a nonnegative integer dependent on the group.

No.	Shorthand	The nilpotency class is defined as ...	The nilpotency class of a nilpotent group $G$ is...
1	upper central series	the length of the upper central series	the smallest nonnegative integer $c$ such that $Z^{c} (G) = G$ . Here, we define $Z^{c} (G)$ inductively as follows: $Z^{c} (G)$ is the inverse image of the center $Z (G / Z^{c - 1} (G))$ under the natural quotient map from $G$ to $G / Z^{c - 1} (G)$ , and $Z^{0} (G)$ is the trivial subgroup.
2	lower central series	the length of the lower central series	the smallest nonnegative integer $c$ such that $[[[. . [G, G], G], G], . . . G]$ is trivial where $G$ is repeated $c + 1$ times. Here, $[,]$ denotes the commutator of two subgroups. In other words, the lower central series of $G$ reaches the identity in finitely many steps.
3	central series	the minimum possible length of a central series	the smallest nonnegative integer $c$ and a chain of subgroups: $G = H_{1} \geq H_{2} \geq \dots \geq H_{c + 1} = {e}$ such that each $H_{i}$ is a normal subgroup of $G$ and $H_{i} / H_{i + 1}$ is in the center of $G / H_{i + 1}$ . In other words, there exists a central series for $G$ of length $c$ .
4	diagonal subnormal in square	the subnormal depth of the diagonal subgroup in the square of the group	the subnormal depth of the subgroup ${(g, g) : g \in G}$ in the square $G \times G$ with subnormal depth $c$ .
5	iterated left-normed commutators trivial	the smallest length such that any iterated left-normed commutator of length more than that becomes trivial	the smallest length $c$ such that any commutator of the form $[[\dots [x_{1}, x_{2}], x_{3}], \dots, x_{c + 1}]$ takes value the identity element, where the $x_{i}$ are (possibly repeated) elements of $G$ .
6	iterated commutators of any form trivial	the smallest length such that any iterated commutator (with any kind of parenthesization of terms) of length more than that becomes trivial	the smallest length $c$ such that any iterated commutator that involves at least $c$ commutator operations (so $c + 1$ original inputs) takes value the identity element. [SHOW MORE] For instance, for $c = 3$ , the expressions $[[x_{1}, x_{2}], [x_{3}, x_{4}]], [[[x_{1}, x_{2}], x_{3}], x_{4}]], [x_{1}, [x_{2}, [x_{3}, x_{4}]]]$ , $[[x_{1}, [x_{2}, x_{3}]], x_{4}], [x_{1}, [[x_{2}, x_{3}], x_{4}]$ all take value the identity element.
7	iterated left-normed commutators trivial (generating set version)	(pick a generating set for the group) the smallest finite length such that any iterated commutator (with any kind of parenthesization of terms) of length more than that becomes trivial	(pick a generating set $S$ for $G$ ) the smallest finite length $c$ such that any iterated commutator that involves at least $c$ commutator operations (so $c + 1$ original inputs) and where the inputs are from $S$ , always takes value the identity element. [SHOW MORE] For instance, for $c = 3$ , the expressions $[[x_{1}, x_{2}], [x_{3}, x_{4}]], [[[x_{1}, x_{2}], x_{3}], x_{4}]], [x_{1}, [x_{2}, [x_{3}, x_{4}]]]$ , $[[x_{1}, [x_{2}, x_{3}]], x_{4}], [x_{1}, [[x_{2}, x_{3}], x_{4}]$ all take value the identity element.

Equivalence of definitions

For full proof, refer: Equivalence of definitions of nilpotency class

Terminology

When we say that a group has nilpotency class $c$ , we usually mean that the nilpotency class of the group is at most equal to $c$ . If we want to say that the class is exactly $c$ , this is stated explicitly.

Particular cases

Value of $c$	Name for groups of nilpotency class at most $c$
0	trivial group only
1	abelian groups (note that nontrivial abelian groups have class exactly 1)
2	group of nilpotency class two. The non-abelian ones among these have class exactly 2
3	group of nilpotency class three

Example

Any group of prime power order is nilpotent. Further information: prime power order implies nilpotent

A group of order $p^{n}$ , with $p$ prime, can have any nilpotency class between 1 and $n - 1$ if $n \geq 2$ . For more information of the number of p-groups of various nilpotency class values for various primes, refer nilpotency class distribution of p-groups.

Relation with other arithmetic functions

Relation with derived length

Further information: Nilpotency class versus derived length

Any nilpotent group is solvable, and there are numerical relations between the nilpotency class and derived length:

Derived length is logarithmically bounded by nilpotency class
Derived length gives no upper bound on nilpotency class: For a derived length greater than $1$ , the value of the derived length gives no upper bound on the value of the nilpotency class.

Other related notions

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+===Equivalent definitions in tabular format===
-For a [[nilpotent group]], the '''nilpotence class''' is defined in any of the following equivalent ways:
+The notion of '''nilpotency class''' (also called '''nilpotence class''') makes sense for any [[nilpotent group]] and is a nonnegative integer dependent on the group.
-* It is the length of the [[upper central series]]
+{| class="sortable" border="1"
-* It is the length of the [[lower central series]]
+! No. !! Shorthand !! The nilpotency class is defined as ... !! The nilpotency class of a nilpotent group <math>G</math> is...
-* It is the length of any central series
+|-
+| 1 || upper central series || the length of the [[upper central series]] || the smallest nonnegative integer <math>c</math> such that <math>Z^c(G) = G</math>. Here, we define <math>Z^c(G)</math> inductively as follows:
+<math>Z^c(G)</math> is the inverse image of the [[defining ingredient::center]] <math>Z(G/Z^{c-1}(G))</math> under the natural [[quotient map]] from <math>G</math> to <math>G/Z^{c-1}(G)</math>, and <math>Z^0(G)</math> is the trivial subgroup.
+|-
+| 2 || lower central series || the length of the [[lower central series]]  || the smallest nonnegative integer <math>c</math> such that <math>[[[..[G,G],G],G],...G]</math> is trivial where <math>G</math> is repeated <math>c + 1</math> times. Here, <math>[,]</math> denotes the [[defining ingredient::commutator of two subgroups]]. In other words, the [[defining ingredient::lower central series]] of <math>G</math> reaches the identity in finitely many steps.
+|-
+| 3 || central series || the minimum possible length of a [[central series]] || the smallest nonnegative integer <math>c</math> and a chain of subgroups: <math>G = H_1 \ge H_2 \ge \dots \ge H_{c+1} = \{ e \}</math> such that each <math>H_i</math> is a [[normal subgroup]] of <math>G</math> and <math>H_i/H_{i+1}</math> is in the center of <math>G/H_{i+1}</math>. In other words, there exists a [[central series]] for <math>G</math> of length <math>c</math>.
+|-
+| 4 || diagonal subnormal in square || the [[defining ingredient::subnormal depth]] of the diagonal subgroup in the [[defining ingredient::square of a group|square]] of the group || the [[subnormal depth]] of the subgroup <math>\{ (g,g) : g \in G \}</math> in the [[square of a group|square]] <math>G \times G</math> with [[subnormal depth]] <math>c</math>.
+|-
+| 5 || iterated left-normed commutators trivial || the smallest length such that any iterated left-normed [[defining ingredient::commutator]] of length more than that becomes trivial || the smallest length <math>c</math> such that any [[defining ingredient::commutator]] of the form <math>[[\dots[x_1,x_2],x_3],\dots,x_{c+1}]</math> takes value the identity element, where the <math>x_i</math> are (possibly repeated) elements of <math>G</math>.
+|-
+| 6 || iterated commutators of any form trivial || the smallest length such that any iterated [[defining ingredient::commutator]] (with any kind of parenthesization of terms) of length more than that becomes trivial || the smallest length <math>c</math> such that any iterated [[defining ingredient::commutator]] that involves at least <math>c</math> commutator operations (so <math>c+1</math> original inputs) takes value the identity element. <toggledisplay>For instance, for <math>c = 3</math>, the expressions <br><math>\! [[x_1,x_2],[x_3,x_4]], [[[x_1,x_2],x_3],x_4]], [x_1,[x_2,[x_3,x_4]]]</math>,<br> <math>[[x_1,[x_2,x_3]],x_4], [x_1,[[x_2,x_3],x_4]</math><br> all take value the identity element.</toggledisplay>
+|-
+| 7 || iterated left-normed commutators trivial (generating set version) || (pick a generating set for the group) the smallest finite length such that any iterated [[defining ingredient::commutator]] (with any kind of parenthesization of terms) of length more than that becomes trivial || (pick a  generating set <math>S</math> for <math>G</math>) the smallest finite  length <math>c</math> such that any iterated [[defining ingredient::commutator]] that involves at least <math>c</math> commutator operations (so <math>c+1</math> original inputs) and where the inputs are from <math>S</math>, always takes value the identity element. <toggledisplay>For instance, for <math>c = 3</math>, the expressions <br><math>\! [[x_1,x_2],[x_3,x_4]], [[[x_1,x_2],x_3],x_4]], [x_1,[x_2,[x_3,x_4]]]</math>,<br> <math>[[x_1,[x_2,x_3]],x_4], [x_1,[[x_2,x_3],x_4]</math><br> all take value the identity element.</toggledisplay>
+|}
+===Equivalence of definitions===
-A group is said to be of class <math>c</math> if its nilpotence class is less than or equal to <math>c</math>.
+{{proofat|[[Equivalence of definitions of nilpotency class]]}}
+===Terminology===
+When we say that a group has nilpotency class <math>c</math>, we usually mean that the nilpotency class of the group is at ''most'' equal to <math>c</math>. If we want to say that the class is exactly <math>c</math>, this is stated explicitly.
+==Particular cases==
+{| class="sortable" border="1"
+! Value of <math>c</math> !! Name for groups of nilpotency class at most <math>c</math>
+|-
+| 0 || [[trivial group]] only
+|-
+| 1 || [[abelian group]]s (note that nontrivial abelian groups have class ''exactly'' 1)
+|-
+| 2 || [[group of nilpotency class two]]. The non-abelian ones among these have class exactly 2
+|-
+| 3 || [[group of nilpotency class three]]
+|}
+==Example==
+Any [[group of prime power order]] is [[nilpotent group|nilpotent]]. {{further|[[prime power order implies nilpotent]]}}
+A group of order <math>p^n</math>, with <math>p</math> prime, can have any nilpotency class between 1 and <math>n - 1</math> if <math>n \ge 2</math>. For more information of the number of p-groups of various nilpotency class values for various primes, refer [[nilpotency class distribution of p-groups]].
+==Relation with other arithmetic functions==
+===Relation with derived length===
+{{further|[[Nilpotency class versus derived length]]}}
+Any nilpotent group is [[solvable group|solvable]], and there are numerical relations between the nilpotency class and derived length:
+* [[Derived length is logarithmically bounded by nilpotency class]]
+* [[Derived length gives no upper bound on nilpotency class]]: For a derived length greater than <math>1</math>, the value of the derived length gives no upper bound on the value of the nilpotency class.
+===Other related notions===
+* [[Exponent-p class]]
+* [[Frattini length]]
+* [[Class row of a polynilpotent group]]