Characteristic direct factor: Difference between revisions

Latest revision as of 02:59, 20 January 2013

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: characteristic subgroup and direct factor
View other subgroup property conjunctions | view all subgroup properties

Definition

A subgroup of a group is termed a characteristic direct factor if it is a characteristic subgroup as well as a direct factor.

Examples

Extreme examples

Every group is a characteristic direct factor of itself.
The trivial subgroup is a characteristic direct factor in any group.

Subgroups satisfying the property

Here are some examples of subgroups in basic/important groups satisfying the property:

Here are some examples of subgroups in relatively less basic/important groups satisfying the property:

Here are some examples of subgroups in even more complicated/less basic groups satisfying the property:

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
transitive subgroup property	Yes	Follows by combining characteristicity is transitive and direct factor is transitive	Suppose $H\leq K\leq G$ are groups such that $H$ is a characteristic direct factor of $K$ and $K$ is a characteristic direct factor of $G$ . Then, $H$ is a characteristic direct factor of $G$ .
trim subgroup property	Yes	Follows from the fact that both the property of being characteristic and the property of being a direct factor satisfy this condition.	In any group $G$ , both the whole group $G$ and the trivial subgroup are characteristic direct factors.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant direct factor	fully invariant subgroup and a direct factor	fully invariant implies characteristic	characteristic direct factor not implies fully invariant	characteristic direct factor\|fully invariant direct factor}}
Hall direct factor	Hall subgroup that is a direct factor	equivalence of definitions of normal Hall subgroup shows that normal Hall subgroups are fully invariant.		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic central factor	characteristic subgroup as well as a central factor	direct factor implies central factor	follows from center is characteristic and the fact that the center is not always a direct factor	\|FULL LIST, MORE INFO
IA-automorphism-invariant direct factor	IA-automorphism-invariant subgroup and a direct factor	follows from characteristic implies IA-automorphism-invariant subgroup
IA-automorphism-balanced subgroup	every IA-automorphism of the whole group restricts to an IA-automorphism of the subgroup.	(via IA-automorphism-invariant direct factor)	(via IA-automorphism-invariant direct factor)	\|FULL LIST, MORE INFO
characteristic subgroup	invariant under all automorphisms			\|FULL LIST, MORE INFO
direct factor	normal subgroup with a normal complement			\|FULL LIST, MORE INFO

Facts

Prime power order implies no proper nontrivial characteristic direct factor

@@ Line 3: / Line 3: @@
 ==Definition==
-===Symbol-free definition===
+A [[subgroup]] of a [[group]] is termed a '''characteristic direct factor''' if it is a [[characteristic subgroup]] as well as a [[direct factor]].
-A [[subgroup]] of a [[group]] is termed a '''characteristic direct factor''' if it is a [[characteristic subgroup]] as well as a [[direct factor]].
+==Examples==
-==Relation with other properties==
+===Extreme examples===
-===Stronger properties===
+* Every group is a characteristic direct factor of itself.
+* The trivial subgroup is a characteristic direct factor in any group.
-* [[Weaker than::Fully invariant direct factor]]: {{proofofstrictimplicationat|[[fully invariant implies characteristic]]|[[characteristic direct factor not implies fully invariant]]}}
+===Subgroups satisfying the property===
-* [[Weaker than::Hall direct factor]]
-===Weaker properties===
+{{subgroups satisfying property conjunction sorted by importance rank|characteristic subgroup|direct factor}}
-* [[Stronger than::Characteristic central factor]]
+==Metaproperties==
-* [[Stronger than::IA-automorphism-invariant direct factor]]
-* [[Stronger than::IA-balanced subgroup]]
-* [[Stronger than::Characteristic subgroup]]
-* [[Stronger than::Direct factor]]
-==Facts==
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::transitive subgroup property]] || Yes || Follows by combining [[characteristicity is transitive]] and [[direct factor is transitive]] || Suppose <math>H \le K \le G</math> are groups such that <math>H</math> is a characteristic direct factor of <math>K</math> and <math>K</math> is a characteristic direct factor of <math>G</math>. Then, <math>H</math> is a characteristic direct factor of <math>G</math>.
+|-
+| [[satisfies metaproperty::trim subgroup property]] || Yes || Follows from the fact that both the property of being characteristic and the property of being a direct factor satisfy this condition. || In any group <math>G</math>, both the whole group <math>G</math> and the trivial subgroup are characteristic direct factors.
+|}
-* [[Prime power order implies no proper nontrivial characteristic direct factor]]
+==Relation with other properties==
-==Metaproperties==
+===Stronger properties===
-{{transitive}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::fully invariant direct factor]] || [[fully invariant subgroup]] and a [[direct factor]] || [[fully invariant implies characteristic]] || [[characteristic direct factor not implies fully invariant]] || {intermediate notions short|characteristic direct factor|fully invariant direct factor}}
+|-
+| [[Weaker than::Hall direct factor]] || [[Hall subgroup]] that is a [[direct factor]] || [[equivalence of definitions of normal Hall subgroup]] shows that normal Hall subgroups are fully invariant. || || {{intermediate notions short|characteristic direct factor|Hall direct factor}}
+|}
-Since both the property of being characteristic and the property of being a direct factor are transitive, so is the property of being a characteristic direct factor.
+===Weaker properties===
-In fact, it is a {{t.i.}}.
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::characteristic central factor]] || [[characteristic subgroup]] as well as a [[central factor]] || [[direct factor implies central factor]] || follows from [[center is characteristic]] and the fact that the center is not always a direct factor || {{intermediate notions short|characteristic central factor|characteristic direct factor}}
+|-
+| [[Stronger than::IA-automorphism-invariant direct factor]] || [[IA-automorphism-invariant subgroup]] and a [[direct factor]] || follows from [[characteristic implies IA-automorphism-invariant subgroup]] || ||
+|-
+| [[Stronger than::IA-automorphism-balanced subgroup]] || every IA-automorphism of the whole group restricts to an IA-automorphism of the subgroup. || (via IA-automorphism-invariant direct factor) || (via IA-automorphism-invariant direct factor) || {{intermediate notions short|IA-automorphism-balanced subgroup|characteristic direct factor}}
+|-
+| [[Stronger than::characteristic subgroup]] || invariant under all automorphisms || || || {{intermediate notions short|characteristic subgroup|characteristic direct factor}}
+|-
+| [[Stronger than::direct factor]] || normal subgroup with a normal complement || || || {{intermediate notions short|direct factor|characteristic direct factor}}
+|}
-{{trim}}
+==Facts==
-Again, since both the property of being characteristic and the property of being a direct factor are trim, so is the property of being a characteristic direct factor.
+* [[Prime power order implies no proper nontrivial characteristic direct factor]]