Normal closure (group theory)

Algebraic structure → Group theory Group theory
Basic notions Subgroup Normal subgroup Quotient group (Semi-)direct product Group homomorphisms kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable action Glossary of group theory List of group theory topics
Finite groups Cyclic group Zn Symmetric group Sn Alternating group An Dihedral group Dn Quaternion group Q Cauchy's theorem Lagrange's theorem Sylow theorems Hall's theorem p-group Elementary abelian group Frobenius group Schur multiplier Classification of finite simple groups cyclic alternating Lie type sporadic
Discrete groups Lattices Integers (${\displaystyle \mathbb {Z} }$) Free group Modular groups PSL(2, ${\displaystyle \mathbb {Z} }$) SL(2, ${\displaystyle \mathbb {Z} }$) Arithmetic group Lattice Hyperbolic group
Topological and Lie groups Solenoid Circle General linear GL(n) Special linear SL(n) Orthogonal O(n) Euclidean E(n) Special orthogonal SO(n) Unitary U(n) Special unitary SU(n) Symplectic Sp(n) G2 F4 E6 E7 E8 Lorentz Poincaré Conformal Diffeomorphism Loop Infinite dimensional Lie group O(∞) SU(∞) Sp(∞)
Algebraic groups Linear algebraic group Reductive group Abelian variety Elliptic curve
v t e

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

This group theory-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e

This page was last edited on 12 August 2023, at 23:33