# T-group (mathematics)

From HandWiki

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

Original source: https://en.wikipedia.org/wiki/T-group (mathematics).
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