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In group theory, the normal closure of a subset S of a group G is the smallest normal subgroup of G containing S.

Properties and description

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

ncl_{G}(S)=\bigcap _{S\subseteq N\triangleleft G}N.

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

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

Therefore one can also write

ncl_{G}(S)=\{g_{1}^{-1}s_{1}^{\epsilon _{1}}g_{1}\dots g_{n}^{-1}s_{n}^{\epsilon _{n}}g_{n}\mid 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 $\varnothing$ is the trivial subgroup.^[2]

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

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.^[3]

Group presentations

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

References

^ ^a ^b Derek F. Holt; Bettina Eick; Eamonn A. O'Brien (2005). Handbook of Computational Group Theory. CRC Press. p. 14. ISBN 1-58488-372-3.
^ Rotman, Joseph J. (1995). An introduction to the theory of groups (Fourth ed.). New York: Springer-Verlag. p. 32. doi:10.1007/978-1-4612-4176-8. ISBN 0-387-94285-8. MR 1307623.
^ 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.
^ 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.

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[HEOB-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 (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.

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@@ Line 13: / Line 13: @@
 <math display=block>ncl_G(S)=\{g_1^{-1}s_1^{\epsilon_1}g_1\dots g_n^{-1}s_n^{\epsilon_n}g_n \mid n\ge 0, \epsilon_i=\pm 1, s_i\in S, g_i \in G\}.</math>
-Any normal subgroup is equal to its normal closure.  The conjugate closure of the [[empty set]] <math>\varnothing</math> is the [[trivial subgroup]].{{cn|date=May 2021}}
+Any normal subgroup is equal to its normal closure.  The conjugate closure of the [[empty set]] <math>\varnothing</math> is the [[trivial subgroup]].<ref>{{cite book |last1=Rotman |first1=Joseph J. |title=An introduction to the theory of groups |date=1995 |publisher=[[Springer-Verlag]] |location=New York |isbn=0-387-94285-8 |page=32 |edition=Fourth |url=https://www.google.com/books/edition/_/7-bBoQEACAAJ?hl=en&sa=X&ved=2ahUKEwid46mc6MvwAhVDEVkFHV_MCBYQre8FMAB6BAgDEDI|MR=1307623|doi=10.1007/978-1-4612-4176-8}}</ref>
 A variety of other notations are used for the normal closure in the literature, including <math>\langle S^G\rangle</math>, <math>\langle S\rangle^G</math>, <math>\langle \langle S\rangle\rangle_G</math>, and <math>\langle\langle S\rangle\rangle^G</math>.