Wielandt subgroup

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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 $W(G)$ of a group $G$ is defined as the intersection, over all $H$ subnormal in $G$, of the groups $N_{G}(H)$.

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.