Smallest normal group containing a set
In
group theory, the normal closure of a
subset
of a
group
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
:
[1]
![{\displaystyle \operatorname {ncl} _{G}(S)=\bigcap _{S\subseteq N\triangleleft G}N.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abd5fc55dc6722bb46e96e347c4ab7c5fc257b5d)
The normal closure
is the smallest normal subgroup of
containing
[1] 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
![{\displaystyle \operatorname {ncl} _{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\dots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}:n\geq 0,\epsilon _{i}=\pm 1,s_{i}\in S,g_{i}\in G\}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8ce0e6ebff5a56703a826263cd8a3eb47d0c947)
Any normal subgroup is equal to its normal closure. The conjugate closure of the
empty set
is the
trivial subgroup.
[2]
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 normal interior or
normal core, defined as the join of all normal subgroups contained in
[3]
Group presentations
For a group
given by a
presentation
with generators
and defining
relators
the presentation notation means that
is the
quotient group
where
is a
free group on
[4]
References