# CN-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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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.

## 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) CA-group|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