Wielandt subgroup
From Groupprops
This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup
View a complete list of subgroup-defining functions OR View a complete list of quotient-defining functions
This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.
VIEW: Definitions built on this | Facts about this: (facts closely related to Wielandt subgroup, all facts related to Wielandt subgroup) |Survey articles about this | Survey articles about definitions built on this
VIEW RELATED: Analogues of this | Variations of this | Opposites of this |
View a list of other standard non-basic definitions
Origin
The notion of Wielandt subgroup was introduced by Wielandt in the paper Ober den Normalizator den Subnormalen Untergruppen, Math Z. 45 (1939), Pages 209-244.
Definition
Symbol-free definition
The Wielandt subgroup of a group is defined as the intersection of the normalizers of all its subnormal subgroups.
Definition with symbols
The Wielandt subgroup of a group is defined as the intersection, over all subnormal in , of the groups .
Relation with other subgroup-defining functions
Smaller subgroup-defining functions
- Baer norm: This is the intersection of normalizers of all subgroups
- Center: This is contained inside the Baer norm as well
Effect of operators
Fixed-point operator
A group equals its Wielandt subgroup if and only if every subnormal subgroup of the group is normal, or equivalently, if and only if the group is a T-group.
References
- Ober den Normalizator den Subnormalen Untergruppen by Wielandt, Math Z. 45 (1939), Pages 209-244.