Marginal subgroup: Difference between revisions

Latest revision as of 19:04, 27 July 2013

Definition

Marginal subgroup for a single word

Suppose $w$ is a word in the letters $x_{1}, x_{2}, \dots, x_{n}$ and $G$ is a group. The marginal subgroup for $w$ in $G$ is the set of all $g \in G$ such that:

$w (x_{1}, x_{2}, \dots, x_{n}) = w (x_{1}, x_{2}, \dots, g x_{i}, x_{i + 1}, \dots x_{n}) = w (x_{1}, x_{2}, \dots, x_{i} g, x_{i + 1}, \dots, x_{n}) \forall x_{1}, x_{2}, \dots, x_{n} \in G, \forall i \in {1, 2, \dots, n}$

That this set is a subgroup is readily verified.

Marginal subgroup for a collection of words

The marginal subgroup for a (possibly infinite) collection of words in a group is the intersection of the marginal subgroups for each of the words in that group.

Marginal subgroup for a variety

The marginal subgroup for a subvariety of the variety of groups is defined in the following equivalent ways:

It is the marginal subgroup for the collection of all words that become trivial in that variety.
It is the marginal subgroup for any collection of words that generates the variety, in the sense that a group is in the variety iff all those words are trivial in it.

Examples

Extreme examples

The whole group is the marginal subgroup for the word $w (x) = x$ .
The trivial subgroup is the marginal subgroup for the empty word, i.e., the word that evaluates to the identity element for any element.

Commutator word and center

Marginal subgroup	Corresponding word or words	Corresponding variety
center	commutator word $[x_{1}, x_{2}] = x_{1} x_{2} x_{1}^{- 1} x_{2}^{- 1}$ (this is the left-normed commutator, but the same holds for the right-normed commutator)	abelian groups
second center	$[[x_{1}, x_{2}], x_{3}]$	groups of nilpotency class at most two
finite $c^{t h}$ member of upper central series	$[\dots [x_{1}, x_{2}], x_{3}], \dots, x_{c + 1}]$	nilpotent groups of class at most $c$
marginal subgroup for variety of metabelian groups	$[[x_{1}, x_{2}], [x_{3}, x_{4}]]$	metabelian groups, i.e., groups of derived length at most two

Power words

The marginal subgroup for the power word $x^{2}$ is the set of central elements of order dividing 2.

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties [SHOW MORE]
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VIEW RELATED: Subgroup property implications | Subgroup property non-implications | Subgroup metaproperty satisfactions | Subgroup metaproperty dissatisfactions | Subgroup property satisfactions |Subgroup property dissatisfactions
VIEW FACTS THAT:use, or start with, a subgroup satisfying the property|prove, or show that, a subgroup satisfies the property

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
strongly intersection-closed subgroup property	Yes	marginality is strongly intersection-closed	If $H_{i}, i \in I$ are all marginal subgroups of $G$ , then so is $⋂_{i \in I} H_{i}$ .
trim subgroup property	Yes		Every group is marginal in itself. The trivial subgroup is marginal in every group.
transitive subgroup property	No	marginality is not transitive	It is possible to have groups $H \leq K \leq G$ such that $H$ is marginal in $K$ and $K$ is marginal in $G$ but $H$ is not marginal in $G$ .
intermediate subgroup condition	No	marginality does not satisfy intermediate subgroup condition	It is possible to have groups $H \leq K \leq G$ such that $H$ is marginal in $G$ but not in $K$ .
direct power-closed subgroup property	Yes	marginality is direct power-closed	Suppose $H$ is a marginal subgroup of a group $G$ , and $α$ is a finite or infinite cardinal. Then, in the direct power $G^{α}$ , $H^{α}$ is a marginal subgroup. In fact, it is marginal for the same variety that $H$ is marginal in $G$ .

Facts

Marginal subgroup is closed in T0 topological group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
marginal subgroup of finite type	we use only finitely many words			\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
weakly marginal subgroup	like marginal subgroup, but we now only require the marginality for a fixed letter of the word rather than for all letters of the word.	direct	weakly marginal not implies marginal	\|FULL LIST, MORE INFO
bound-word subgroup	(see definition)
algebraic subgroup	algebraic subset, i.e., intersection of elementary algebraic subsets			\|FULL LIST, MORE INFO
unconditionally closed subgroup		(via algebraic)	(via algebraic)	\|FULL LIST, MORE INFO
strictly characteristic subgroup	invariant under all surjective endomorphisms
characteristic subgroup	invariant under all automorphisms			\|FULL LIST, MORE INFO
direct power-closed characteristic subgroup	in any direct power of the group, the corresponding direct power of the subgroup is characteristic			\|FULL LIST, MORE INFO
finite direct power-closed characteristic subgroup	in any finite direct power of the group, the corresponding direct power of the subgroup is characteristic			\|FULL LIST, MORE INFO

Dual property

The notion of marginal subgroup is somewhat dual to the notion of verbal subgroup, which is the subgroup generated by all elements realized using the given word.

@@ Line 5: / Line 5: @@
 Suppose <math>w</math> is a [[word]] in the letters <math>x_1,x_2,\dots,x_n</math> and <math>G</math> is a [[group]]. The '''marginal subgroup''' for <math>w</math> in <math>G</math> is the set of all <math>g \in G</math> such that:
-<math>\! w(x_1,x_2,\dots,x_n) = w(x_1,x_2,\dots,gx_i,x_{i+1}, \dots x_n) = w(x_1,x_2,\dots,x_ig,x_{i+1},\dots,x_n) \ \forall \ x_1,x_2,\dots,x_n \in G</math>
+<math>\! w(x_1,x_2,\dots,x_n) = w(x_1,x_2,\dots,gx_i,x_{i+1}, \dots x_n) = w(x_1,x_2,\dots,x_ig,x_{i+1},\dots,x_n) \ \forall \ x_1,x_2,\dots,x_n \in G, \forall i \in \{ 1,2,\dots,n \}</math>
 That this set is a subgroup is readily verified.
@@ Line 12: / Line 12: @@
 The marginal subgroup for a (possibly infinite) collection of words in a group is the intersection of the marginal subgroups for each of the words in that group.
+===Marginal subgroup for a variety===
+The marginal subgroup for a subvariety of the variety of groups is defined in the following equivalent ways:
+# It is the marginal subgroup for the collection of ''all'' words that become trivial in that variety.
+# It is the marginal subgroup for any collection of words that ''generates'' the variety, in the sense that a group is in the variety iff all those words are trivial in it.
 ==Examples==
@@ Line 22: / Line 29: @@
 ===Commutator word and center===
-* The marginal subgroup for the [[commutator]] word <math>[x_1,x_2] = x_1x_2x_1^{-1}x_2^{-1}</math> (this is the left-normed commutator, but the same holds for the right-normed commutator) is precisely the [[center]].
+{| class="sortable" border="1"
-* The marginal subgroup for a left-normed iterated commutator such as <math>[\dots [x_1,x_2],x_3],\dots,x_n]</math> is the <math>(n-1)^{th}</math> member of the [[upper central series]]. (This is not completely obvious for <math>n \ge 3</math>).
+! Marginal subgroup !! Corresponding word or words !! Corresponding variety
+|-
+| [[center]] || [[commutator]] word <math>[x_1,x_2] = x_1x_2x_1^{-1}x_2^{-1}</math> (this is the left-normed commutator, but the same holds for the right-normed commutator) ||[[abelian group]]s
+|-
+| [[second center]] || <math>[[x_1,x_2],x_3]</math> || [[group of nilpotency class two|groups of nilpotency class at most two]]
+|-
+| finite <math>c^{th}</math> member of [[upper central series]] || <math>[\dots [x_1,x_2],x_3],\dots,x_{c+1}]</math> || [[nilpotent group]]s of class at most <math>c</math>
+|-
+| [[marginal subgroup for variety of metabelian groups]] || <math>[[x_1,x_2],[x_3,x_4]]</math> || [[metabelian group]]s, i.e., groups of derived length at most two
+|}
 ===Power words===
@@ Line 29: / Line 45: @@
 {{subgroup property}}
 ==Metaproperties==
@@ Line 34: / Line 51: @@
 ! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
 |-
-| [[quotient-transitive subgroup property]] || ? || ? || ?
+| [[satisfies metaproperty::strongly intersection-closed subgroup property]] || Yes || [[marginality is strongly intersection-closed]] || If <math>H_i, i \in I</math> are all marginal subgroups of <math>G</math>, then so is <math>\bigcap_{i \in I} H_i</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || ||Every group is marginal in itself. The trivial subgroup is marginal in every group.
+|-
+| [[dissatisfies metaproperty::transitive subgroup property]] || No || [[marginality is not transitive]] || It is possible to have groups <math>H \le K \le G</matH> such that <math>H</math> is marginal in <math>K</math> and <math>K</matH> is marginal in <math>G</math> but <math>H</math> is not marginal in <math>G</math>.
+|-
+| [[dissatisfies metaproperty::intermediate subgroup condition]] || No || [[marginality does not satisfy intermediate subgroup condition]] || It is possible to have groups <math>H \le K \le G</matH> such that <math>H</math> is marginal in <math>G</math> but not in <math>K</math>.
+|-
+| [[satisfies metaproperty::direct power-closed subgroup property]] || Yes || [[marginality is direct power-closed]] || Suppose <math>H</math> is a marginal subgroup of a group <math>G</math>, and <math>\alpha</math> is a finite or infinite cardinal. Then, in the [[direct power]] <math>G^\alpha</math>, <math>H^\alpha</math> is a marginal subgroup. In fact, it is marginal for the same variety that <math>H</math> is marginal in <math>G</math>.
 |}
+==Facts==
+* [[Marginal subgroup is closed in T0 topological group]]
 ==Relation with other properties==
@@ Line 43: / Line 73: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| member of [[upper central series]] (finite part) || || || ||
+| [[Weaker than::marginal subgroup of finite type]] || we use only finitely many words || || || {{intermediate notions short|marginal subgroup|marginal subgroup of finite type}}
 |}
@@ Line 51: / Line 81: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::bound-word subgroup]] || can be expressed in terms of words and equations || || ||
+| [[Stronger than::weakly marginal subgroup]] || like marginal subgroup, but we now only require the marginality for a fixed letter of the word rather than for all letters of the word. || direct || [[weakly marginal not implies marginal]] || {{intermediate notions short|weakly marginal subgroup|marginal subgroup}}
+|-
+| [[Stronger than::bound-word subgroup]] || (see definition) || || ||
+|-
+| [[Stronger than::algebraic subgroup]] || [[algebraic subset]], i.e., intersection of [[elementary algebraic subset]]s || || || {{intermediate notions short|algebraic subgroup|marginal subgroup}}
+|-
+| [[Stronger than::unconditionally closed subgroup]] || || (via algebraic) || (via algebraic) || {{intermediate notions short|unconditionally closed subgroup|marginal subgroup}}
 |-
 | [[Stronger than::strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || || ||
@@ Line 57: / Line 93: @@
 | [[Stronger than::characteristic subgroup]] || invariant under all [[automorphism]]s || || || {{intermediate notions short|characteristic subgroup|marginal subgroup}}
 |-
-| [[Stronger than::finite direct power-closed characteristic subgroup]] || in any finite direct power of the group, the corresponding direct power of the subgroup is characteristic || || ||
+| [[Stronger than::direct power-closed characteristic subgroup]] || in any direct power of the group, the corresponding direct power of the subgroup is characteristic || || || {{intermediate notions short|characteristic subgroup|direct power-closed characteristic subgroup}}
+|-
+| [[Stronger than::finite direct power-closed characteristic subgroup]]||  in any finite direct power of the group, the corresponding direct power of the subgroup is characteristic || || || {{intermediate notions short|characteristic subgroup|finite direct power-closed characteristic subgroup}}
 |}