CN-group
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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
Contents
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 ![]() |
---|---|---|---|
1 | element centralizers are nilpotent | the centralizer of any non-identity element is a nilpotent subgroup. | for every non-identity element ![]() ![]() ![]() ![]() ![]() |
2 | subgroup centralizers are nilpotent | the centralizer of any nontrivial subgroup is a nilpotent subgroup. | for every nontrivial subgroup ![]() ![]() ![]() ![]() |
3 | subgroups: nilpotent or centerless | every subgroup of the group is either a nilpotent group or a centerless group. | for every nontrivial subgroup ![]() ![]() ![]() ![]() ![]() |
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 ![]() ![]() ![]() ![]() |
finite direct product-closed group property | Yes | Suppose ![]() ![]() ![]() |
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) | 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 |