Group satisfying normalizer condition: Difference between revisions

Latest revision as of 04:03, 17 April 2017

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Definition

A group $G$ is said to satisfy the normalizer condition, if it satisfies the following equivalent conditions:

The normalizer $N_{G}(H)$ of any proper subgroup $H$ properly contains it
There is no proper self-normalizing subgroup of $G$
Every subgroup of $G$ is ascendant

Groups satisfying the normalizer condition have been termed N-groups but the term N-group is also used for groups with a particular condition on normalizers of solvable subgroups.

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes		If $G$ is a group satisfying normalizer condition, and $H$ is a subgroup of $G$ , then $H$ also satisfies normalizer condition.
quotient-closed group property	Yes		If $G$ is a group satisfying normalizer condition, and $H$ is a normal subgroup of $G$ , then the quotient group $G/H$ also satisfies normalizer condition.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
nilpotent group		nilpotent implies normalizer condition	normalizer condition not implies nilpotent	\|FULL LIST, MORE INFO
group in which every subgroup is subnormal				\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Gruenberg group	every cyclic subgroup is ascendant			\|FULL LIST, MORE INFO
locally nilpotent group	every finitely generated subgroup is nilpotent	notmalizer condition implies locally nilpotent	locally nilpotent not implies normalizer condition	\|FULL LIST, MORE INFO
group having no proper abnormal subgroup	there is no proper abnormal subgroup			\|FULL LIST, MORE INFO
group in which every maximal subgroup is normal	every maximal subgroup is a normal subgroup			\|FULL LIST, MORE INFO

Metaproperties

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@@ Line 5: / Line 5: @@
 ==Definition==
-===Symbol-free definition===
+A [[group]] <math>G</math> is said to satisfy the '''normalizer condition''', if it satisfies the following equivalent conditions:
-A [[group]] is said to satisfy the '''normalizer condition''', if it satisfies the following equivalent conditions:
+# The [[normalizer]] <math>N_G(H)</math> of any [[proper subgroup]] <math>H</math> properly contains it
+# There is no [[proper subgroup|proper]] [[self-normalizing subgroup]] of <math>G</math>
+# Every subgroup of <math>G</math> is [[ascendant subgroup|ascendant]]
-* The [[normalizer]] of any [[proper subgroup]] properly contains it
+Groups satisfying the normalizer condition have been termed '''N-groups''' but the term [[N-group]] is also used for groups with a particular condition on normalizers of solvable subgroups.
-* There is no [[proper subgroup|proper]] [[self-normalizing subgroup]]
-* Every subgroup is [[ascendant subgroup|ascendant]]
-===Definition with symbols===
+== Metaproperties ==
-A [[group]] <math>G</math> is said to satisfy a '''normalizer condition''' if for any proper subgroup <math>H</math> of <math>G</math>, <math>H < N_G(H)</math> with the inclusion being strict (that is, <math>H</math> is ''properly contained'' in its [[normalizer]]).
-Groups satisfying the normalizer condition have been termed '''N-groups''' but the term [[N-group]] is also used for groups with a particular condition on normalizers of solvable subgroups.
+{| class="sortable" border="1"
+! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
+|-
+| [[satisfies metaproperty::subgroup-closed group property]] || Yes || || If <math>G</math> is a group satisfying normalizer condition, and <math>H</math> is a subgroup of <math>G</math>, then <math>H</math> also satisfies normalizer condition.
+|-
+| [[satisfies metaproperty::quotient-closed group property]] || Yes || || If <math>G</math> is a group satisfying normalizer condition, and <math>H</math> is a normal subgroup of <math>G</math>, then the quotient group <math>G/H</math> also satisfies normalizer condition.
+|}
 ==Relation with other properties==
 ===Stronger properties===
-* [[Weaker than::Nilpotent group]]: It turns out that for a [[finitely generated group]], the two properties are equivalent. {{proofofstrictimplicationat|[[Nilpotent implies normalizer condition]]|[[Normalizer condition not implies nilpotent]]}}
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::nilpotent group]] || || [[nilpotent implies normalizer condition]] || [[normalizer condition not implies nilpotent]] || {{intermediate notions short|group satisfying normalizer condition|nilpotent group}}
+|-
+| [[Weaker than::group in which every subgroup is subnormal]] || || || || {{intermediate notions short|group satisfying normalizer condition|group in which every subgroup is subnormal}}
+|}
 ===Weaker properties===
-* [[Stronger than::Locally nilpotent group]]
-* [[Stronger than::Group in which every maximal subgroup is normal]]
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::Gruenberg group]] || every cyclic subgroup is ascendant || || || {{intermediate notions short|Gruenberg group|group satisfying normalizer condition}}
+|-
+| [[Stronger than::locally nilpotent group]] || every finitely generated subgroup is nilpotent || [[notmalizer condition implies locally nilpotent]] || [[locally nilpotent not implies normalizer condition]] || {{intermediate notions short|locally nilpotent group|group satisfying normalizer condition}}
+|-
+| [[Stronger than::group having no proper abnormal subgroup]] || there is no proper [[abnormal subgroup]] || || || {{intermediate notions short|group having no proper abnormal subgroup|group satisfying normalizer condition}}
+|-
+| [[Stronger than::group in which every maximal subgroup is normal]] || every [[maximal subgroup]] is a [[normal subgroup]] || || || {{intermediate notions short|group in which every maximal subgroup is normal|group satisfying normalizer condition}}
+|}
 ==Metaproperties==