T-group (mathematics)
In mathematics, in the field of group theory, a T-group is a group in which the property of normality is transitive, that is, every subnormal subgroup is normal. Here are some facts about T-groups:
- Every simple group is a T-group.
- Every quasisimple group is a T-group.
- Every abelian group is a T-group.
- Every Hamiltonian group is a T-group.
- Every nilpotent T-group is either abelian or Hamiltonian, because in a nilpotent group, every subgroup is subnormal.
- Every normal subgroup of a T-group is a T-group.
- Every homomorphic image of a T-group is a T-group.
- Every solvable T-group is metabelian.
The solvable T-groups were characterized by Wolfgang Gaschütz as being exactly the solvable groups G with an abelian normal Hall subgroup H of odd order such that the quotient group G/H is a Dedekind group and H is acted upon by conjugation as a group of power automorphisms by G.
A PT-group is a group in which permutability is transitive. A finite T-group is a PT-group.
References
- Robinson, Derek J.S. (1996), A Course in the Theory of Groups, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94461-6
- Ballester-Bolinches, Adolfo; Esteban-Romero, Ramon; Asaad, Mohamed (2010), Products of Finite Groups, Walter de Gruyter, ISBN 978-3-11-022061-2