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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Contents
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 is termed a T-group if ... |
---|---|---|---|
1 | subnormal equals normal | any subnormal subgroup of the group is normal in the group. | for any subgroup such that there exists an ascending chain where each is normal in , we have that is normal in . |
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 such that is normal in and is normal in , we have that is normal in . |
3 | normal equals transitively normal | any normal subgroup is transitively normal | for any normal subgroup of , it is true that any normal subgroup of is normal in . |
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) |
Symmetric group:S6 | 720 (763) |
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 and a subgroup of such that 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 is T-group and is a normal subgroup of . Then, the quotient group is also a T-group. |
normal subgroup-closed group property | Yes | T-group property is normal subgroup-closed. | Suppose is T-group and is a normal subgroup of . Then, 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