T-group

Origin
Ernest Best and Olga Taussky defined these groups in their paper A class of Groups, published in 1942. They named them t-groups.

Wolfgang Gaschütz described the soluble T-groups in 1957 as the groups G in which the nilpotent residual is an abelian Hall subgroup L of odd order such that G/L is Dedekind and G normalizes every subgroup of L.

The notion of T-group was discussed by Derek J.S. Robinson in his paper A Note on Finite Groups in which normality is transitive published in 1968.

Definition
A group is termed a T-group if it satisfies the following equivalent conditions:

Extreme examples

 * The trivial group is a T-group.

Formalisms
The property of being a T-group can be viewed as any of these subgroup property collapses:


 * The collapse transitively normal subgroup = normal subgroup
 * The collapse normal subgroup = defining ingredient::subnormal subgroup

Conjunction with other properties

 * nilpotent T-group which is the same as Dedekind group

Weaker properties

 * HN-group (when we are working with finite groups)

Subgroup-closure
The property of being a group such that every subgroup of it is a T-group, is termed the property of being a T*-group.

Testing
There is no built-in GAP command to check whether a group is a T-group, but a short snippet of code, available at GAP:IsTGroup, can achieve this.

Textbook references

 * , Page 402-405, Section 13.4: Groups in which normality is a transitive relation

Journal references

 * A Note on Finite Groups in which normality is transitive by Derek J.S. Robinson, Proceedings of the Americal Mathematical Society Vol. 19 No. 4, Aug 1968, pages 933-937
 * Gruppen, in denen das Normalteilersein transitiv ist by Wolfgang Gaschütz, J. reine angew. Math., 198, 1957, pages 87-92
 * A class of groups by Ernest Best and Olga Taussky, Proc. Irish. Acad., 47, 1942, pages 55-62