Centralizer-annihilating endomorphism-invariant subgroup: Difference between revisions

Latest revision as of 22:00, 16 February 2013

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Definition

A subgroup $H$ of a group $G$ is termed a centralizer-annihilating endomorphism-invariant subgroup of $G$ if, for every centralizer-annihilating endomorphism $σ$ of $G$ , $σ (H)$ is contained in $H$ . Here, a centralizer-annihilating endomorphism of $G$ with respect to $H$ is an endomorphism whose kernel contains the centralizer $C_{G} (H)$ .

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup				\|FULL LIST, MORE INFO
fully normalized potentially fully invariant subgroup		fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant

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 ===Stronger properties===
-{| class="wikitable" border="1"
+{| class="sortable" border="1"
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
 | [[Weaker than::fully invariant subgroup]] || || || || {{intermediate notions short|centralizer-annihilating endomorphism-invariant subgroup|fully invariant subgroup}}
 |-
-| [[Weaker than::fully normalized potentially fully invariant subgroup]] || || [fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant]] || ||
+| [[Weaker than::fully normalized potentially fully invariant subgroup]] || || [[fully normalized and potentially fully invariant implies centralizer-annihilating endomorphism-invariant]] || ||
 |}