In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, 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, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
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 iden ...
is said to be fully normalized
if every automorphism of the subgroup lifts to an
inner automorphism
In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra
Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within thos ...
of the whole group. Another way of putting this is that the natural embedding from the
Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
of the subgroup to its
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
is surjective.
In symbols, a subgroup
is fully normalized in
if, given an automorphism
of
, there is a
such that the map
, when restricted to
is equal to
.
Some facts:
* Every group can be embedded as a
normal and fully normalized subgroup of a bigger group. A natural construction for this is the
holomorph, which is its
semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. It is usually denoted with the symbol . There are two closely related concepts of semidirect product:
* an ''inner'' sem ...
with its automorphism group.
* A
complete group is fully normalized in any bigger group in which it is embedded because every automorphism of it is inner.
* Every fully normalized subgroup has the
automorphism extension property.
References
Subgroup properties
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