Normal closure (group theory)

This article is about the normal closure of a subset of a group. For the normal closure of a field extension, see Normal closure (field theory).

In group theory, the normal closure of a subset ${\displaystyle S}$ of a group ${\displaystyle G}$ is the smallest normal subgroup of ${\displaystyle G}$ containing ${\displaystyle S.}$

Properties and description

Formally, if ${\displaystyle G}$ is a group and ${\displaystyle S}$ is a subset of ${\displaystyle G,}$ the normal closure ${\displaystyle \operatorname {ncl} _{G}(S)}$ of ${\displaystyle S}$ is the intersection of all normal subgroups of ${\displaystyle G}$ containing ${\displaystyle S}$: ^[1]

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

The normal closure ${\displaystyle \operatorname {ncl} _{G}(S)}$ is the smallest normal subgroup of ${\displaystyle G}$ containing ${\displaystyle S,}$ ^[1] in the sense that ${\displaystyle \operatorname {ncl} _{G}(S)}$ is a subset of every normal subgroup of ${\displaystyle G}$ that contains ${\displaystyle S.}$

The subgroup ${\displaystyle \operatorname {ncl} _{G}(S)}$ is generated by the set ${\displaystyle S^{G}=\{s^{g}:g\in G\}=\{g^{-1}sg:g\in G\}}$ of all conjugates of elements of ${\displaystyle S}$ in ${\displaystyle G.}$

Therefore one can also write

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

Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set ${\displaystyle \varnothing }$ is the trivial subgroup. ^[2]

A variety of other notations are used for the normal closure in the literature, including ${\displaystyle \langle S^{G}\rangle ,}$${\displaystyle \langle S\rangle ^{G},}$${\displaystyle \langle \langle S\rangle \rangle _{G},}$ and ${\displaystyle \langle \langle S\rangle \rangle ^{G}.}$

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 ${\displaystyle S.}$ ^[3]

Group presentations

For a group ${\displaystyle G}$ given by a presentation ${\displaystyle G=\langle S\mid R\rangle }$ with generators ${\displaystyle S}$ and defining relators ${\displaystyle R,}$ the presentation notation means that ${\displaystyle G}$ is the quotient group ${\displaystyle G=F(S)/\operatorname {ncl} _{F(S)}(R),}$ where ${\displaystyle F(S)}$ is a free group on ${\displaystyle S.}$ ^[4]

References

1. Derek F. Holt; Bettina Eick; Eamonn A. O'Brien (2005). Handbook of Computational Group Theory. CRC Press. p.  14. ISBN   1-58488-372-3.
2. Rotman, Joseph J. (1995). An introduction to the theory of groups. Graduate Texts in Mathematics. Vol. 148 (Fourth ed.). New York: Springer-Verlag. p. 32. doi:10.1007/978-1-4612-4176-8. ISBN   0-387-94285-8. MR   1307623.
3. Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. Vol. 80 (2nd ed.). Springer-Verlag. p. 16. ISBN   0-387-94461-3. Zbl   0836.20001.
4. Lyndon, Roger C.; Schupp, Paul E. (2001). Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin. p. 87. ISBN   3-540-41158-5. MR   1812024.