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This article defines a group property: a property that can be evaluated to true/false for any given group
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RANDOM GROUP PROPERTY: Noetherian group: A group where every subgroup is finitely generated.

Contents 1 History 1.1 Origin 2 Definition 2.1 Symbol-free definition 2.2 Definition with symbols 3 Formalisms 3.1 In terms of the subgroup property collapse operator 3.2 In terms of the transitivity-forcing operator 4 Relation with other properties 4.1 Stronger properties 4.2 Conjunction with other properties 4.3 Weaker properties 5 Metaproperties 5.1 Subgroups 5.2 Quotients 5.3 Normal subgroups 6 Effect of property operators 6.1 Subgroup-closure 7 Testing 7.1 GAP code 8 References 8.1 Textbook references 8.2 Journal references 9 External links

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 it satisfies the following equivalent conditions:

• Any subnormal subgroup of the group is normal in the group.
• Any 2-subnormal subgroup (i.e. any normal subgroup of a normal subgroup) of a normal subgroup of it is normal in it
• Any normal subgroup is transitively normal

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 (normal subgroup) satisfies the second property (transitively normal subgroup), and vice versa.
View other group properties obtained in this way

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

• Abelian group
• Dedekind group
• T*-group
• Simple group

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
View a complete list of quotient-closed group properties

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.

Testing

GAP code

One can write code to test this group property in GAP (Groups, Algorithms and Programming), though there is no direct command for it.
View the GAP code for testing this group property at: IsTGroup
View other GAP-codable group properties | View group properties with in-built commands

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.

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