Group satisfying normalizer condition: Difference between revisions

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Definition

Symbol-free 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

Relation with other properties

Stronger properties

Nilpotent group: It turns out that for a finitely generated group, the two properties are equivalent. For proof of the implication, refer Nilpotent implies normalizer condition and for proof of its strictness (i.e. the reverse implication being false) refer Normalizer condition not implies nilpotent.
Group in which every subgroup is subnormal

Weaker properties

Metaproperties

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@@ Line 7: / Line 7: @@
 ===Symbol-free definition===
-A [[group]] is said to satisfy the '''normalizer condition''', if it satisfies the following equivalent conditions:
+A [[group]] <math>G</math> is said to satisfy the '''normalizer condition''', if it satisfies the following equivalent conditions:
-* The [[normalizer]] of any [[proper subgroup]] properly contains it
+* 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]]
+* There is no [[proper subgroup|proper]] [[self-normalizing subgroup]] of <math>G</math>
-* Every subgroup is [[ascendant subgroup|ascendant]]
+* Every subgroup of <math>G</math> is [[ascendant subgroup|ascendant]]
-===Definition with symbols===
-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.
 ==Relation with other properties==