Weakly normal-homomorph-containing subgroup: Difference between revisions

Latest revision as of 02:53, 7 February 2010

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Definition

Definition with symbols

A subgroup $N$ of a group $G$ is termed a weakly normal-homomorph-containing subgroup if $N$ is a normal subgroup of $G$ and the following holds:

Suppose $φ : N \to G$ is a homomorphism of groups such that for any normal subgroup $H$ of $G$ contained in $N$ , we have $φ (H)$ is normal in $G$ . Then, $φ (N) \leq N$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normal-homomorph-containing subgroup	contains any homomorphic image that is normal in whole group	normal-homomorph-containing implies weakly normal-homomorph-containing		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Normality-preserving endomorphism-invariant subgroup	invariant under all normality-preserving endomorphisms	weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant	normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing	\|FULL LIST, MORE INFO
Strictly characteristic subgroup	invariant under all surjective endomorphisms	Weakly normal-homomorph-containing implies strictly characteristic	Strictly characteristic not implies weakly normal-homomorph-containing	\|FULL LIST, MORE INFO
Characteristic subgroup	invariant under all automorphisms	(via strictly characteristic)	(via strictly characteristic)	\|FULL LIST, MORE INFO
Normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	\|FULL LIST, MORE INFO

@@ Line 14: / Line 14: @@
 ===Stronger properties===
-* [[Weaker than::Normal-homomorph-containing subgroup]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure)  !! Intermediate notions
+|-
+| [[Weaker than::Normal-homomorph-containing subgroup]] || contains any homomorphic image that is normal in whole group || [[normal-homomorph-containing implies weakly normal-homomorph-containing]] || || {{intermediate notions short|weakly normal-homomorph-containing subgroup|normal-homomorph-containing subgroup}}
+|}
 ===Weaker properties===
-* [[Stronger than::Strictly characteristic subgroup]]: {{proofat|[[Weakly normal-homomorph-containing implies strictly characteristic]]}}
+{| class="sortable" border="1"
-* [[Stronger than::Characteristic subgroup]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure)  !! Intermediate notions
+|-
+| [[Stronger than::Normality-preserving endomorphism-invariant subgroup]] || invariant under all [[normality-preserving endomorphism]]s || [[weakly normal-homomorph-containing implies normality-preserving endomorphism-invariant]] || [[normality-preserving endomorphism-invariant not implies weakly normal-homomorph-containing]] || {{intermediate notions short|normality-preserving endomorphism-invariant subgroup|weakly normal-homomorph-containing subgroup}}
+|-
+| [[Stronger than::Strictly characteristic subgroup]] || invariant under all [[surjective endomorphism]]s || [[Weakly normal-homomorph-containing implies strictly characteristic]] || [[Strictly characteristic not implies weakly normal-homomorph-containing]] || {{intermediate notions short|strictly characteristic subgroup|weakly normal-homomorph-containing subgroup}}
+|-
+| [[Stronger than::Characteristic subgroup]] || invariant under all [[automorphism]]s || (via strictly characteristic) || (via strictly characteristic) || {{intermediate notions short|characteristic subgroup|weakly normal-homomorph-containing subgroup}}
+|-
+| [[Stronger than::Normal subgroup]] || invariant under all [[inner automorphism]]s || (via characteristic) || (via characteristic) || {{intermediate notions short|normal subgroup|weakly normal-homomorph-containing subgroup}}
+|}