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

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

No.	Shorthand	A group is termed a T-group if ...	A group $G$ is termed a T-group if ...
1	subnormal equals normal	any subnormal subgroup of the group is normal in the group.	for any subgroup $H \le G$ such that there exists an ascending chain $H = H_0 \le H_1 \le H_2 \le \dots \le H_n = G$ where each $H_i$ is normal in $H_{i+1}$, we have that $H$ is normal in $G$.
2	2-subnormal equals normal	any 2-subnormal subgroup (i.e. any normal subgroup of a normal subgroup) is normal in the group.	for any subgroups $H \le K \le G$ such that $H$ is normal in $K$ and $K$ is normal in $G$, we have that $H$ is normal in $G$.
3	normal equals transitively normal	any normal subgroup is transitively normal	for any normal subgroup $K$ of $G$, it is true that any normal subgroup of $K$ is normal in $G$.

This definition is presented using a tabular format. |View all pages with definitions in tabular format

Examples

Extreme examples

• The trivial group is a T-group.

Groups satisfying the property

Here are some basic/important groups satisfying the property:

	GAP ID
Cyclic group:Z2	2 (1)
Cyclic group:Z3	3 (1)
Cyclic group:Z4	4 (1)
Symmetric group:S3	6 (1)

Here are some relatively less basic/important groups satisfying the property:

	GAP ID
Alternating group:A5	60 (5)
Alternating group:A6	360 (118)
Direct product of Z4 and Z2	8 (2)
Projective special linear group:PSL(3,2)	168 (42)
Quaternion group	8 (4)
Special linear group:SL(2,5)	120 (5)
Symmetric group:S5	120 (34)

Here are some even more complicated/less basic groups satisfying the property:

	GAP ID
Alternating group:A7
Special linear group:SL(2,7)	336 (114)
Special linear group:SL(2,9)	720 (409)

Groups dissatisfying the property

Here are some basic/important groups that do not satisfy the property:

Here are some relatively less basic/important groups that do not satisfy the property:

	GAP ID
Alternating group:A4	12 (3)
Dihedral group:D8	8 (3)
Special linear group:SL(2,3)	24 (3)
Symmetric group:S4	24 (12)

Here are some even more complicated/less basic groups that do not satisfy the property:

	GAP ID
Binary octahedral group	48 (28)
Central product of D8 and Z4	16 (13)
Dihedral group:D16	16 (7)
Direct product of D8 and Z2	16 (11)
General linear group:GL(2,3)	48 (29)
Mathieu group:M9	72 (41)
Nontrivial semidirect product of Z4 and Z4	16 (4)
Semidihedral group:SD16	16 (8)

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subgroup-closed group property	No	T-group property is not subgroup-closed	It is possible to have a T-group $G$ and a subgroup $H$ of $G$ such that $H$ is not a T-group. (A T-group for which every subgroup is a T-group is termed a T*-group.
quotient-closed group property	Yes	T-group property is quotient-closed	Suppose $G$ is T-group and $H$ is a normal subgroup of $G$. Then, the quotient group $G/H$ is also a T-group.
normal subgroup-closed group property	Yes	T-group property is normal subgroup-closed.	Suppose $G$ is T-group and $H$ is a normal subgroup of $G$. Then, $H$ is also a T-group.

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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Dedekind group	every subgroup is normal	(direct from definition)	simple non-abelian groups provide counterexamples	Group in which every subgroup is pronormal, T*-group|FULL LIST, MORE INFO
abelian group	any two elements commute	(via Dedekind)	(via Dedekind)	Group in which every normal subgroup is a central factor, Group in which every subgroup is pronormal, T*-group|FULL LIST, MORE INFO
T*-group	every subgroup is a T-group	by definition	T-group property is not subgroup-closed	|FULL LIST, MORE INFO
simple group	no proper nontrivial group	by definition	any abelian non-simple group gives a counterexample	Group in which every normal subgroup is a direct factor|FULL LIST, MORE INFO

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)

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