Normal closure (group theory)

In group theory, the normal closure of a subset

is the smallest normal subgroup of

the normal closure

ncl

is the intersection of all normal subgroups of

ncl

{\displaystyle \operatorname {ncl} _{G}(S)=\bigcap _{S\subseteq N\triangleleft G}N.}

The normal closure

ncl

{\displaystyle \operatorname {ncl} _{G}(S)}

is the smallest normal subgroup of

ncl

{\displaystyle \operatorname {ncl} _{G}(S)}

is a subset of every normal subgroup of

ncl

{\displaystyle \operatorname {ncl} _{G}(S)}

is generated by the set

of all conjugates of elements of

ncl

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

and defining relators

the presentation notation means that

is the quotient group

is a free group on

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