In
mathematics, in the field of
group theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups.
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...
, a T-group is a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
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
SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service.
The d ...
is a T-group.
*Every
quasisimple group
In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension ''E'' of a simple group ''S''. In other words, there is a short exact sequence
:1 \to Z(E) \to E \to S \to 1
such that E = , E/ ...
is a T-group.
*Every
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
is a T-group.
*Every
Hamiltonian group In group theory, a Dedekind group is a group ''G'' such that every subgroup of ''G'' is normal.
All abelian groups are Dedekind groups.
A non-abelian Dedekind group is called a Hamiltonian group.
The most familiar (and smallest) example of a Hamilt ...
is a T-group.
*Every
nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0.
The term was introduced by Benjamin Peirce in the context of his work on the cla ...
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
In mathematics, specifically group theory, a Hall subgroup of a finite group ''G'' is a subgroup whose order is coprime to its index. They were introduced by the group theorist .
Definitions
A Hall divisor (also called a unitary divisor) o ...
''H'' of odd
order such that the
quotient group ''G''/''H'' is a
Dedekind group and ''H'' is
acted upon by
conjugation
Conjugation or conjugate may refer to:
Linguistics
* Grammatical conjugation, the modification of a verb from its basic form
* Emotive conjugation or Russell's conjugation, the use of loaded language
Mathematics
* Complex conjugation, the chang ...
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
*
*
{{DEFAULTSORT:T-Group (Mathematics)
Properties of groups