Maximal among abelian normal implies self-centralizing in nilpotent

From Groupprops
Jump to: navigation, search
This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property must also satisfy the second group property
View all group property implications | View all group property non-implications
|
Property "Page" (as page type) with input value "{{{stronger}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
Property "Page" (as page type) with input value "{{{weaker}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.
This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a nilpotent group. That is, it states that in a Nilpotent group (?), every subgroup satisfying the first subgroup property (i.e., Maximal among Abelian normal subgroups (?)) must also satisfy the second subgroup property (i.e., Self-centralizing subgroup (?)). In other words, every maximal among Abelian normal subgroups of nilpotent group is a self-centralizing subgroup of nilpotent group.
View all subgroup property implications in nilpotent groups | View all subgroup property non-implications in nilpotent groups | View all subgroup property implications | View all subgroup property non-implications
This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being self-centralizing. In other words, the centralizer of the subgroup in the group is contained in the subgroup
View other similar statements

Statement

Property-theoretic statement

In terms of group properties: the group property of being a nilpotent group is stronger than, or implies, the property of being a group in which maximal among Abelian normal implies self-centralizing.

In terms of subgroup properties: when restricting to the class of nilpotent groups, the subgroup property of being maximal among Abelian normal subgroups implies the subgroup property of being a self-centralizing subgroup.

Statement with symbols

Suppose G is a nilpotent group and H is maximal among Abelian normal subgroups of G: in other words, H is an Abelian normal subgroup such that there is no Abelian normal subgroup of G containing H. Then, H is a self-centralizing subgroup, i.e., C_G(H) = H.

Related facts

Analogues in other algebraic structures

Facts used

  1. Normality is centralizer-closed
  2. Normality satisfies image condition
  3. Nilpotence is quotient-closed: Any quotient of a nilpotent group is nilpotent.
  4. Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
  5. Cyclic over central implies Abelian
  6. Normality satisfies inverse image condition