CN-group

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This group property is obtained by applying the condition that the centralizer of every element satisfies the following group property: Abelian group

History

Origin

The concept and terminology of CN-groups was introduced by Michio Suzuki in his attempts to solve Burnside's conjecture (which later became the odd-order theorem).

Definition

Below are listed some equivalent definitions of CN-group, also called centralizer is nilpotent group:

No.	Shorthand	A group is termed a CN-group if ...	A group $G$ is termed a CN-group if ...
1	element centralizers are nilpotent	the centralizer of any non-identity element is a nilpotent subgroup.	for every non-identity element $x\in G$, the centralizer $C_{G}(x)$ (i.e., the set of all elements of $G$ that commute with $x$) is a nilpotent subgroup of $G$.
2	subgroup centralizers are nilpotent	the centralizer of any nontrivial subgroup is a nilpotent subgroup.	for every nontrivial subgroup $H$ of $G$, the centralizer $C_{G}(H)$ is a nilpotent subgroup of $G$.
3	subgroups: nilpotent or centerless	every subgroup of the group is either a nilpotent group or a centerless group.	for every nontrivial subgroup $H$ of $G$, either $H$ is nilpotent or $H$ is centerless, i.e., the center of $H$ is trivial.

Equivalence of definitions

Further information: equivalence of definitions of CN-group

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	Yes	direct from version (3) of the definition	Suppose $G$ is a CN-group and $H$ is a subgroup of $G$. Then, $H$ is also a CN-group.
finite direct product-closed group property	Yes		Suppose $G_{1}$ and $G_{2}$ are CN-groups. Then, the external direct product $G_{1}\times G_{2}$ is also a CN-group.

Facts

• For finite CN-groups: Sylow subgroups for distinct primes in CN-group centralize each other iff they have non-identity elements that centralize each other
• Quotient group of finite CN-group by solvable normal subgroup is CN-group

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
abelian group	the whole group is abelian	(via nilpotent)		|FULL LIST, MORE INFO
nilpotent group	the whole group is nilpotent	(obvious)	CN not implies nilpotent	|FULL LIST, MORE INFO
CA-group	the centralizer of every non-identity element is abelian	follows from abelian implies nilpotent	any nilpotent non-abelian group, use a central element	|FULL LIST, MORE INFO