CN-group

From Groupprops
Jump to: navigation, search
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
View a complete list of group properties
VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

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

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