Normal closure (group theory)

In group theory, the normal closure of a subset

S

{\displaystyle S}

of a group

G

{\displaystyle G}

is the smallest normal subgroup of

G

{\displaystyle G}

containing

S

.

{\displaystyle S.}

Properties and description

Formally, if

G

{\displaystyle G}

is a group and

S

{\displaystyle S}

is a subset of

G

,

{\displaystyle G,}

the normal closure

ncl

G

⁡

(

S

)

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

of

S

{\displaystyle S}

is the intersection of all normal subgroups of

G

{\displaystyle G}

containing

S

{\displaystyle S}

:

ncl

G

⁡

(

S

)

=

⋂

S

⊆

N

◃

G

N

.

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

The normal closure

ncl

G

⁡

(

S

)

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

is the smallest normal subgroup of

G

{\displaystyle G}

containing

S

,

{\displaystyle S,}

in the sense that

ncl

G

⁡

(

S

)

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

is a subset of every normal subgroup of

G

{\displaystyle G}

that contains

S

.

{\displaystyle S.}

The subgroup

ncl

G

⁡

(

S

)

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

is the subgroup generated by the set

S

G

=

{

s

g

:

s

∈

S

,

g

∈

G

}

=

{

g

−

1

s

g

:

s

∈

S

,

g

∈

G

}

{\displaystyle S^{G}=\{s^{g}:s\in S,g\in G\}=\{g^{-1}sg:s\in S,g\in G\}}

of all conjugates of elements of

S

{\displaystyle S}

in

G

.

{\displaystyle G.}

Therefore, one can also write the subgroup as the set of all products of conjugates of elements of

S

{\displaystyle S}

or their inverses:

ncl

G

⁡

(

S

)

=

{

g

1

−

1

s

1

ϵ

1

g

1

⋯

g

n

−

1

s

n

ϵ

n

g

n

:

n

≥

0

,

ϵ

i

=

±

1

,

s

i

∈

S

,

g

i

∈

G

}

.

{\displaystyle \operatorname {ncl} _{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\cdots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}:n\geq 0,\epsilon _{i}=\pm 1,s_{i}\in S,g_{i}\in G\}.}

Any normal subgroup is equal to its normal closure. The normal closure of the empty set

∅

{\displaystyle \varnothing }

is the trivial subgroup.

A variety of other notations are used for the normal closure in the literature, including

⟨

S

G

⟩

,

{\displaystyle \langle S^{G}\rangle ,}

⟨

S

⟩

G

,

{\displaystyle \langle S\rangle ^{G},}

⟨

⟨

S

⟩

⟩

G

,

{\displaystyle \langle \langle S\rangle \rangle _{G},}

and

⟨

⟨

S

⟩

⟩

G

.

{\displaystyle \langle \langle S\rangle \rangle ^{G}.}

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

S

.

{\displaystyle S.}

Group presentations

For a group

G

{\displaystyle G}

given by a presentation

G

=

⟨

S

∣

R

⟩

{\displaystyle G=\langle S\mid R\rangle }

with generators

S

{\displaystyle S}

and defining relators

R

,

{\displaystyle R,}

the presentation notation means that

G

{\displaystyle G}

is the quotient group

G

=

F

(

S

)

/

ncl

F

(

S

)

⁡

(

R

)

,

{\displaystyle G=F(S)/\operatorname {ncl} _{F(S)}(R),}

where

F

(

S

)

{\displaystyle F(S)}

is a free group on

S

.