Locally finite group: Difference between revisions

Latest revision as of 14:14, 11 May 2017

This is a variation of finiteness (groups)|Find other variations of finiteness (groups) |

Definition

A group $G$ is said to be locally finite if it satisfies the following equivalent conditions:

Every subgroup of it that is finitely generated is in fact finite. In other words, for any positive integer $n$ and elements $g_{1}, g_{2}, \dots, g_{n} \in G$ , the subgroup $⟨ g_{1}, g_{2}, \dots, g_{n} ⟩$ is a finite group.
It is the direct limit of a directed system of finite groups. In other words, there exists $G_{i}, i \in I$ , a directed system of finite groups, such that $G$ is isomorphic to the direct limit of the $G_{i}$ s.
It is the direct limit of a directed system of finite groups, where all the maps are injective. In other words, there exists $G_{i}, i \in I$ , a directed system of finite groups with all maps injective, such that $G$ is isomorphic to the direct limit of the $G_{i}$ s.
If $H$ is a finite subgroup of $G$ and $x \in G$ , then the subgroup $⟨ H, x ⟩$ is also a finite subgroup of $G$ .

Examples

Every finite group is locally finite.
For a prime number $p$ , the $p$ -quasicyclic group is a locally finite group. It is obtained as a direct limit of inclusions:

$0 \to Z / p Z \to Z / p^{2} Z \to \dots \to Z / p^{n} Z \to$

where the inclusion maps are multiplication by $p$ maps. Equivalently, it can be thought of as the multiplicative group of the union of all $(p^{n})^{t h}$ roots of unity in the complex numbers for all $n$ .

The group of rational numbers modulo integers $Q / Z$ is an example of a locally finite group.
The finitary symmetric group on a possibly infinite set is locally finite, because any finite subset of the group has finite support and hence lives inside the symmetric group on a finite subset.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	local finiteness is subgroup-closed	If $G$ is a locally finite group, and $H$ is a subgroup of $G$ , then $H$ is also locally finite.
quotient-closed group property	Yes	local finiteness is quotient-closed	If $G$ is a locally finite group, and $H$ is a normal subgroup of $G$ , then the quotient group $G / H$ is also locally finite.
extension-closed group property	Yes	local finiteness is extension-closed	If $G$ is a group and $H$ is a normal subgroup such that both $H$ and $G / H$ are locally finite, then $G$ is also locally finite.
restricted direct product-closed group property	Yes	local finiteness is restricted direct product-closed	If $G_{i}, i \in I$ are all locally finite groups, so is the restricted external direct product of the $G_{i}$ s.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
finite group	the group itself is finite, i.e., has finitely many elements		locally finite not implies finite (or, see the example in the #Examples section)	\|FULL LIST, MORE INFO
group embeddable in a finitary symmetric group	the group is isomorphic to a subgroup of a finitary symmetric group		locally finite not implies embeddable in finitary symmetric group	\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
periodic group	every element has finite order	locally finite implies periodic	periodic not implies locally finite	\|FULL LIST, MORE INFO
group generated by periodic elements	the group has a generating set comprising elements all of which have finite order.			\|FULL LIST, MORE INFO
2-locally finite group	any subgroup generated by two elements is finite		follows from Golod's theorem on locally finite groups	\|FULL LIST, MORE INFO

Conjunction with other properties

Conjunction	Other component of conjunction	Comment
periodic abelian group	abelian group	for an abelian group, being periodic is equivalent to being locally finite. See equivalence of definitions of periodic abelian group for more.
periodic nilpotent group	nilpotent group	for a nilpotent group, being periodic is equivalent to being locally finite. See equivalence of definitions of periodic nilpotent group for more.
periodic solvable group	solvable group	for a solvable group, being periodic is equivalent to being locally finite. See equivalence of definitions of periodic solvable group for more.
locally finite simple group	simple group	finitary alternating groups are examples.

Opposite properties

Finitely generated group: A group that is both finitely generated and locally finite must be finite.

Dual properties

Residually finite group
Profinite group: A topological, completed version of residually finite

Formalisms

BEWARE! This section of the article uses terminology local to the wiki, possibly without giving a full explanation of the terminology used (though efforts have been made to clarify terminology as much as possible within the particular context)

In terms of the locally operator

This property is obtained by applying the locally operator to the property: finite group
View other properties obtained by applying the locally operator

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+A [[group]] <math>G</math> is said to be '''locally finite''' if it satisfies the following equivalent conditions:
-A [[group]] is said to be '''locally finite''' if it satisfies the following equivalent conditions:
+# Every [[subgroup]] of it that is [[finitely generated group|finitely generated]] is in fact [[finite group|finite]]. In other words, for any positive integer <math>n</math> and elements <math>g_1,g_2,\dots,g_n \in G</math>, the subgroup <math>\langle g_1, g_2, \dots, g_n \rangle</math> is a finite group.
+# It is the [[direct limit]] of a directed system of finite groups. In other words, there exists <math>G_i, i \in I</math>, a directed system of finite groups, such that <math>G</math> is isomorphic to the direct limit of the <math>G_i</math>s.
-# Every [[subgroup]] of it that is [[finitely generated group|finitely generated]], is in fact [[finite group|finite]].
+# It is the [[direct limit]] of a directed system of finite groups, where all the maps are injective. In other words, there exists <math>G_i, i \in I</math>, a directed system of finite groups with all maps injective, such that <math>G</math> is isomorphic to the direct limit of the <math>G_i</math>s.
-# It is the [[direct limit]] of a directed system of finite groups.
+# If <math>H</math> is a [[finite subgroup]] of <math>G</math> and <math>x \in G</math>, then the subgroup <math>\langle H, x \rangle</math> is also a finite subgroup of <math>G</math>.
-===Definition with symbols===
-A [[group]] <math>G</math> is said to be '''locally finite''' if for any finite subset <math>g_1, g_2, \ldots, g_n \in G</math> the group generated by the <math>g_i</math>s is a [[finite group]].
 ==Examples==
+* Every finite group is locally finite.
 * For a [[prime number]] <math>p</math>, the <math>p</math>-[[quasicyclic group]] is a locally finite group. It is obtained as a direct limit of inclusions:
@@ Line 21: / Line 18: @@
 where the inclusion maps are multiplication by <math>p</math> maps. Equivalently, it can be thought of as the multiplicative group of the union of all <math>(p^n)^{th}</math> roots of unity in the complex numbers for all <math>n</math>.
+* The [[group of rational numbers modulo integers]] <math>\mathbb{Q}/\mathbb{Z}</math> is an example of a locally finite group.
 * The [[finitary symmetric group]] on a possibly infinite set is locally finite, because any finite subset of the group has finite support and hence lives inside the symmetric group on a finite subset.
+==Metaproperties==
-==Formalisms==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
-{{wikilocal-section}}
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || [[local finiteness is subgroup-closed]] || If <math>G</math> is a locally finite group, and <math>H</math> is a [[subgroup]] of <math>G</math>, then <math>H</math> is also locally finite.
-{{obtainedbyapplyingthe|locally operator|finite group}}
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || [[local finiteness is quotient-closed]] || If <math>G</math> is a locally finite group, and <math>H</math> is a [[normal subgroup]] of <math>G</math>, then the [[quotient group]] <math>G/H</math> is also locally finite.
+|-
+| [[satisfies metaproperty::extension-closed group property]] || Yes || [[local finiteness is extension-closed]] || If <math>G</math> is a group and <math>H</math> is a [[normal subgroup]] such that both <math>H</math> and <math>G/H</math> are locally finite, then <math>G</math> is also locally finite.
+|-
+| [[satisfies metaproperty::restricted direct product-closed group property]] || Yes || [[local finiteness is restricted direct product-closed]] || If <math>G_i, i \in I</math> are all locally finite groups, so is the [[restricted external direct product]] of the <math>G_i</math>s.
+|}
 ==Relation with other properties==
@@ Line 34: / Line 39: @@
 ===Stronger properties===
-* [[Weaker than::Finite group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::finite group]] || the group itself is finite, i.e., has finitely many elements || || [[locally finite not implies finite]] (or, see the example in the [[#Examples]] section) || {{intermediate notions short|locally finite group|finite group}}
+|-
+| [[Weaker than::group embeddable in a finitary symmetric group]] || the group is isomorphic to a subgroup of a [[finitary symmetric group]] || || [[locally finite not implies embeddable in finitary symmetric group]] || {{intermediate notions short|locally finite group|group embeddable in a finitary symmetric group}}
+|}
 ===Weaker properties===
-* [[Stronger than::Periodic group]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::periodic group]] || every element has finite order || [[locally finite implies periodic]] || [[periodic not implies locally finite]] || {{intermediate notions short|periodic group|locally finite group}}
+|-
+| [[Stronger than::group generated by periodic elements]] || the group has a [[generating set of a group|generating set]] comprising elements all of which have finite order. || || || {{intermediate notions short|group generated by periodic elements|locally finite group}}
+|-
+| [[Stronger than::2-locally finite group]] || any subgroup generated by two elements is finite || || follows from [[Golod's theorem on locally finite groups]] || {{intermediate notions short|2-locally finite group|locally finite group}}
+|}
+===Conjunction with other properties===
+{| class="sortable" border="1"
+! Conjunction !! Other component of conjunction !! Comment
+|-
+| [[Weaker than::periodic abelian group]] || [[abelian group]] || for an abelian group, being periodic is equivalent to being locally finite. See [[equivalence of definitions of periodic abelian group]] for more.
+|-
+| [[Weaker than::periodic nilpotent group]] || [[nilpotent group]] || for a nilpotent group, being periodic is equivalent to being locally finite. See [[equivalence of definitions of periodic nilpotent group]] for more.
+|-
+| [[Weaker than::periodic solvable group]] || [[solvable group]] || for a solvable group, being periodic is equivalent to being locally finite. See [[equivalence of definitions of periodic solvable group]] for more.
+|-
+| [[Weaker than::locally finite simple group]] || [[simple group]] || [[finitary alternating group]]s are examples.
+|}
 ===Opposite properties===
@@ Line 49: / Line 82: @@
 * [[Profinite group]]: A topological, ''completed'' version of residually finite
-==Metaproperties==
+==Formalisms==
-{{S-closed}}
+{{wikilocal-section}}
-Any subgroup of a locally finite group is locally finite.
+{{obtainedbyapplyingthe|locally operator|finite group}}
-{{Q-closed}}
-Any quotient of a locally finite group is locally finite.
-{{restricted DP-closed}}
-A restricted direct product of locally finite groups is locally finite. In particular, a direct product of finitely many locally finite groups is locally finite.
-{{proofat|[[Local finiteness is restricted direct product-closed]]}}