In
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 ...
, the normal closure of a
subset 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 the smallest
normal subgroup of
containing
Properties and description
Formally, if
is a group and
is a subset of
the normal closure
of
is the intersection of all normal subgroups of
containing
:
The normal closure
is the smallest normal subgroup of
containing
in the sense that
is a subset of every normal subgroup of
that contains
The subgroup
is
generated by the set
of all
conjugates of elements of
in
Therefore one can also write
Any normal subgroup is equal to its normal closure. The conjugate closure of the
empty set is the
trivial subgroup
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually ...
.
A variety of other notations are used for the normal closure in the literature, including
and
Dual to the concept of normal closure is that of or , defined as the join of all normal subgroups contained in
Group presentations
For a group
given by a
presentation
A presentation conveys information from a speaker to an audience. Presentations are typically demonstrations, introduction, lecture, or speech meant to inform, persuade, inspire, motivate, build goodwill, or present a new idea/product. Presenta ...
with generators
and defining
relators
the presentation notation means that
is the
quotient group where
is a
free group
In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
on
[
]
References
Group theory
Closure operators
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