Maximal among abelian normal implies self-centralizing in nilpotent
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
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
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 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 self-centralizing subgroup, i.e., .
- Maximal among abelian normal implies self-centralizing in supersolvable: The corresponding statement holds for supersolvable groups.
- Maximal among abelian normal not implies self-centralizing in solvable: The corresponding statement does not hold for solvable groups.
- Maximal among abelian characteristic not implies self-centralizing 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
- Normality is centralizer-closed
- Normality satisfies image condition
- Nilpotence is quotient-closed: 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