Maximal among abelian normal implies selfcentralizing in nilpotent
From Groupprops
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
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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., Selfcentralizing subgroup (?)). In other words, every maximal among Abelian normal subgroups of nilpotent group is a selfcentralizing subgroup of nilpotent group.
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This article gives the statement, and possibly proof, of a particular subgroup of kind of subgroup in a group being selfcentralizing. In other words, the centralizer of the subgroup in the group is contained in the subgroup
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Contents
Statement
Propertytheoretic 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 selfcentralizing.
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 selfcentralizing subgroup.
Statement with symbols
Suppose is a nilpotent group and is maximal among Abelian normal subgroups of : in other words, is an Abelian normal subgroup such that there is no Abelian normal subgroup of containing . Then, is a selfcentralizing subgroup, i.e., .
Related facts
 Maximal among abelian normal implies selfcentralizing in supersolvable: The corresponding statement holds for supersolvable groups.
 Maximal among abelian normal not implies selfcentralizing in solvable: The corresponding statement does not hold for solvable groups.
 Maximal among abelian characteristic not implies selfcentralizing in nilpotent: The corresponding statement does not hold when normality is replaced by characteristicity.
 Thompson's critical subgroup theorem: A similar result that uses a subgroup that is maximal among Abelian characteristic subgroups.
Analogues in other algebraic structures
Facts used
 Normality is centralizerclosed
 Normality satisfies image condition
 Nilpotence is quotientclosed: Any quotient of a nilpotent group is nilpotent.
 Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup
 Cyclic over central implies Abelian
 Normality satisfies inverse image condition