Quotient-subisomorph-containing subgroup: Difference between revisions

Latest revision as of 21:23, 26 July 2013

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Definition

The following are some equivalent definitions of quotient-subisomorph-containing subgroup.

No.	Shorthand	A subgroup $H$ of a group $G$ is termed a quasiverbal subgroup if ...
1	contained in the kernel of any homomorphism to the quotient	for any homomorphism of groups $φ : G \to G / H$ , $H$ is contained in the kernel of $φ$ .
2	intersection of normal subgroups whose quotients satisfy a subgroup-closed group property	there exists a subgroup-closed group property $α$ such that $H$ is the intersection of all normal subgroups $N$ of $G$ for which the quotient group $G / N$ satisfies property $α$ .
3	smallest normal subgroup whose quotient satisfies a subgroup-closed group property	there exists a subgroup-closed group property $α$ such that $H$ is the unique smallest normal subgroup of $G$ for which the quotient group $G / H$ satisfies $α$ . Note that every $α$ that works for the preceding definition need not work for this definition.

Relation with other properties

Stronger properties

Property	Meaning	Proof of strictness (reverse implication failure)	Intermediate notions
verbal subgroup	union of the images of word maps		\|FULL LIST, MORE INFO
pseudoverbal subgroup	intersection of normal subgroups with quotients in a pseudovariety		\|FULL LIST, MORE INFO
quasiverbal subgroup	intersection of normal subgroups with quotients in a quasivariety		\|FULL LIST, MORE INFO
normal subgroup having no nontrivial homomorphism to its quotient group	no nontrivial homomorphism from the subgroup to the quotient group	\|FULL LIST, MORE INFO
normal Sylow subgroup	normal and a Sylow subgroup		\|FULL LIST, MORE INFO
Normal Hall subgroup	normal and a Hall subgroup		quotient-subisomorph-containing subgroup\|normal Hall subgroup}}

Weaker properties

Related properties

Homomorph-containing subgroup

@@ Line 4: / Line 4: @@
 ==Definition==
-===Definition with symbols===
+The following are some equivalent definitions of '''quotient-subisomorph-containing subgroup'''.
+{| class="sortable" border="1"
+! No. !! Shorthand !! A subgroup <math>H</math> of a group <math>G</math> is termed a quasiverbal subgroup if ...
+|-
+| 1 || contained in the kernel of any homomorphism to the quotient || for any [[homomorphism of groups]] <math>\varphi:G \to G/H</math>, <math>H</math> is contained in the kernel of <math>\varphi</math>.
+|-
+| 2 || intersection of normal subgroups whose quotients satisfy a subgroup-closed group property || there exists a [[subgroup-closed group property]] <math>\alpha</math> such that <math>H</math> is the intersection of all [[normal subgroup]]s <math>N</math> of <math>G</math> for which the [[quotient group]] <math>G/N</math> satisfies property <math>\alpha</math>.
+|-
+| 3 || smallest normal subgroup whose quotient satisfies a subgroup-closed group property || there exists a [[subgroup-closed group property]] <math>\alpha</math> such that <math>H</math> is the unique smallest [[normal subgroup]] of <math>G</math> for which the quotient group <math>G/H</math> satisfies <math>\alpha</math>. Note that every <math>\alpha</math> that works for the preceding definition need not work for this definition.
+|}
-A [[subgroup]] <math>H</math> of a [[group]] <math>G</math> is termed '''quotient-homomorph-containing''' if <math>H</math> is a [[defining ingredient::normal subgroup]] of <math>G</math>, and for any <math>\varphi \in \operatorname{Hom}(G,G/H)</math>, <math>H \le \operatorname{Ker}\varphi</math>.
 ==Relation with other properties==
@@ Line 12: / Line 21: @@
 ===Stronger properties===
-* [[Weaker than::Verbal subgroup]]
+{| class="sortable" border="1"
-* [[Weaker than::Normal subgroup having no nontrivial homomorphism to its quotient group]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Weaker than::Normal Sylow subgroup]]
+|-
-* [[Weaker than::Normal Hall subgroup]]
+| [[Weaker than::verbal subgroup]] || union of the images of [[word map]]s || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|verbal subgroup}}
+|-
+| [[Weaker than::pseudoverbal subgroup]] || intersection of normal subgroups with quotients in a pseudovariety || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|pseudoverbal subgroup}}
+|-
+| [[Weaker than::quasiverbal subgroup]] || intersection of normal subgroups with quotients in a quasivariety || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|quasiverbal subgroup}}
+|-
+| [[Weaker than::normal subgroup having no nontrivial homomorphism to its quotient group]] || no nontrivial homomorphism from the subgroup to the quotient group || || {{intermediate notions short|quotient-subisomorph-containing subgroup|normal subgroup having no nontrivial homomorphism to its quotient group}}
+|-
+| [[Weaker than::normal Sylow subgroup]] || normal and a [[Sylow subgroup]] || || || {{intermediate notions short|quotient-subisomorph-containing subgroup|normal Sylow subgroup}}
+|-
+| [[Weaker than::Normal Hall subgroup]] || normal and a [[Hall subgroup]] || || || {intermediate notions short|quotient-subisomorph-containing subgroup|normal Hall subgroup}}
+|}
 ===Weaker properties===
 * [[Stronger than::Weakly image-closed fully invariant subgroup]]
-* [[Stronger than::Fully characteristic subgroup]]
+* [[Stronger than::Fully invariant subgroup]]
 * [[Stronger than::Strictly characteristic subgroup]]
 * [[Stronger than::Characteristic subgroup]]