Normal-homomorph-containing subgroup: Difference between revisions

Latest revision as of 17:38, 29 May 2009

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Definition

A subgroup $N$ of a group $G$ is termed normal-homomorph-containing if $N$ is a normal subgroup of $G$ , and for any homomorphism $φ : N \to G$ such that $φ (N)$ is also a normal subgroup of $G$ , we have $φ (N) \leq N$ .

Relation with other properties

Stronger properties

Weaker properties

Weakly normal-homomorph-containing subgroup
Strictly characteristic subgroup: For proof of the implication, refer Normal-homomorph-containing implies strictly characteristic and for proof of its strictness (i.e. the reverse implication being false) refer Strictly characteristic not implies normal-homomorph-containing. Also related:
- Characteristic subgroup
- Normal subgroup
Normal-isomorph-containing subgroup

@@ Line 11: / Line 11: @@
 * [[Weaker than::Homomorph-containing subgroup]]: Also related:
+** [[Weaker than::Subhomomorph-containing subgroup]]
 ** [[Weaker than::Order-containing subgroup]]
 ** [[Weaker than::Variety-containing subgroup]]
 ** [[Weaker than::Normal Sylow subgroup]]
 ** [[Weaker than::Normal Hall subgroup]]
+* [[Weaker than::Normal-subhomomorph-containing subgroup]]
 ===Weaker properties===
-* [[Stronger than::Strictly characteristic subgroup]]: {{proofat|[[Normal-homomorph-containing implies strictly characteristic]]}}
+* [[Stronger than::Weakly normal-homomorph-containing subgroup]]
+* [[Stronger than::Strictly characteristic subgroup]]: {{proofofstrictimplicationat|[[Normal-homomorph-containing implies strictly characteristic]]|[[Strictly characteristic not implies normal-homomorph-containing]]}} Also related:
+** [[Stronger than::Characteristic subgroup]]
+** [[Stronger than::Normal subgroup]]
 * [[Stronger than::Normal-isomorph-containing subgroup]]
-* [[Stronger than::Characteristic subgroup]]