“Transitively normal subgroup”的意思、由来-开放百科全书

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Transitively normal subgroup

释义

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In mathematics, in the field of group theory, a subgroup of a group is said to be transitively normal in the group if every normal subgroup of the subgroup is also normal in the whole group. In symbols, is a transitively normal subgroup of if for every normal in , we have that is normal in .

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 normal subgroup 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.

See also