In group theory, the normal closure of a subset
is the smallest normal subgroup of
containing
is a group and
the normal closure
is the intersection of all normal subgroups of
The normal closure
is the smallest normal subgroup of
[1] in the sense that
is a subset of every normal subgroup of
The subgroup
is generated by the set
of all conjugates of elements of
ϵ
ϵ
ϵ
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
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] For a group
given by a presentation
with generators
and defining relators
the presentation notation means that
is the quotient group
is a free group on
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