T-group

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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History

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

Symbol-free definition

A group is termed a T-group if any subnormal subgroup of the group is normal in the group. In other words, a T group is a group such that any normal subgroup of a normal subgroup of it is normal in it.

Definition with symbols

A group $G$ is termed a T-group if whenever $H$ is normal in $G$ and $K$ is normal in $H$ , $K$ is also normal in $G$ .

Formalisms

In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property ({{{1}}}Property "Defining ingredient" (as page type) with input value "{{{1}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.) satisfies the second property ({{{2}}}Property "Defining ingredient" (as page type) with input value "{{{2}}}" contains invalid characters or is incomplete and therefore can cause unexpected results during a query or annotation process.), and vice versa.
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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 = subnormal subgroup

In terms of the transitivity-forcing operator

This property is obtained by applying the transitivity-forcing operator to the property: normal subgroup
View other properties obtained by applying the transitivity-forcing operator

Relation with other properties

Stronger properties

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)

Metaproperties

Subgroups

This group property is not subgroup-closed, viz., we can have a group satisfying the property, with a subgroup not satisfying the property

A subgroup of a T-group need not be a T-group. A group in which every subgroup is a T-group, is termed a T*-group.

Quotients

This group property is quotient-closed, viz., any quotient of a group satisfying the property also has the property
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Any quotient of a T-group is a T-group. This follows from the fact that a subgroup in the quotient is normal if and only if its full inverse image is normal in the original group.

Normal subgroups

This group property is normal subgroup-closed: any normal subgroup (and hence, any subnormal subgroup) of a group with the property, also has the property
View other normal subgroup-closed group properties

Any normal subgroup (and more generally, any subnormal subgroup) of a T-group is a T-group. Further information: T is normal subgroup-closed

Effect of property operators

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.

References

Textbook references

A Course in the Theory of Groups by Derek J. S. Robinson, ISBN 0387944613^{More info}, 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

External links

JSTOR link for Robinson's paper: Access restricted to subscribers