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

subgroup In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...

of 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 ...

is said to be transitively normal in the group if every

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...

of the subgroup is also normal in the whole group. In symbols,

H

is a transitively normal subgroup of

G

if for every

K

normal in

H

, we have that

K

is normal in

G

. An alternate way to characterize these subgroups is: every ''normal subgroup preserving automorphism'' of the whole group must restrict to a ''normal subgroup preserving automorphism'' of the subgroup. Here are some facts about transitively normal subgroups: *Every

of a transitively normal subgroup is normal. *Every direct factor, or more generally, every central factor is transitively normal. Thus, every central subgroup is transitively normal. *A transitively normal subgroup of a transitively normal subgroup is transitively normal. *A transitively normal subgroup is normal.

References

See also