“T-group (mathematics)”的意思、由来-开放百科全书

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T-group (mathematics)

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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

{{Citation | last1=Robinson | first1=Derek J.S. | title=A Course in the Theory of Groups | publisher=Springer-Verlag | location=Berlin, New York | isbn=978-0-387-94461-6 | year=1996}}
{{citation|first1=Adolfo|last1=Ballester-Bolinches|first2=Ramon|last2=Esteban-Romero|first3=Mohamed|last3=Asaad|title=Products of Finite Groups|year=2010|publisher=Walter de Gruyter|isbn=978-3-11-022061-2}}

1 : Properties of groups

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