Normal closure (group theory)
| Algebraic structure → Group theory Group theory |
|---|
 |
In group theory, the normal closure of a subset [math]\displaystyle{ S }[/math] of a group [math]\displaystyle{ G }[/math] is the smallest normal subgroup of [math]\displaystyle{ G }[/math] containing [math]\displaystyle{ S. }[/math]
Properties and description
Formally, if [math]\displaystyle{ G }[/math] is a group and [math]\displaystyle{ S }[/math] is a subset of [math]\displaystyle{ G, }[/math] the normal closure [math]\displaystyle{ \operatorname{ncl}_G(S) }[/math] of [math]\displaystyle{ S }[/math] is the intersection of all normal subgroups of [math]\displaystyle{ G }[/math] containing [math]\displaystyle{ S }[/math]:[1] [math]\displaystyle{ \operatorname{ncl}_G(S) = \bigcap_{S \subseteq N \triangleleft G} N. }[/math]
The normal closure [math]\displaystyle{ \operatorname{ncl}_G(S) }[/math] is the smallest normal subgroup of [math]\displaystyle{ G }[/math] containing [math]\displaystyle{ S, }[/math][1] in the sense that [math]\displaystyle{ \operatorname{ncl}_G(S) }[/math] is a subset of every normal subgroup of [math]\displaystyle{ G }[/math] that contains [math]\displaystyle{ S. }[/math]
The subgroup [math]\displaystyle{ \operatorname{ncl}_G(S) }[/math] is generated by the set [math]\displaystyle{ S^G=\{s^g : g\in G\} = \{g^{-1}sg : g\in G\} }[/math] of all conjugates of elements of [math]\displaystyle{ S }[/math] in [math]\displaystyle{ G. }[/math]
Therefore one can also write [math]\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\}. }[/math]
Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set [math]\displaystyle{ \varnothing }[/math] is the trivial subgroup.[2]
A variety of other notations are used for the normal closure in the literature, including [math]\displaystyle{ \langle S^G\rangle, }[/math] [math]\displaystyle{ \langle S\rangle^G, }[/math] [math]\displaystyle{ \langle \langle S\rangle\rangle_G, }[/math] and [math]\displaystyle{ \langle\langle S\rangle\rangle^G. }[/math]
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 [math]\displaystyle{ S. }[/math][3]
Group presentations
For a group [math]\displaystyle{ G }[/math] given by a presentation [math]\displaystyle{ G=\langle S \mid R\rangle }[/math] with generators [math]\displaystyle{ S }[/math] and defining relators [math]\displaystyle{ R, }[/math] the presentation notation means that [math]\displaystyle{ G }[/math] is the quotient group [math]\displaystyle{ G = F(S) / \operatorname{ncl}_{F(S)}(R), }[/math] where [math]\displaystyle{ F(S) }[/math] is a free group on [math]\displaystyle{ S. }[/math][4]
References
- ↑ 1.0 1.1 Derek F. Holt; Bettina Eick; Eamonn A. O'Brien (2005). Handbook of Computational Group Theory. CRC Press. p. 14. ISBN 1-58488-372-3. https://archive.org/details/handbookofcomput0000holt/page/14.
- ↑ Rotman, Joseph J. (1995). An introduction to the theory of groups. Graduate Texts in Mathematics. 148 (Fourth ed.). New York: Springer-Verlag. p. 32. doi:10.1007/978-1-4612-4176-8. ISBN 0-387-94285-8. https://books.google.com/books?id=7-bBoQEACAAJ.
- ↑ Robinson, Derek J. S. (1996). A Course in the Theory of Groups. Graduate Texts in Mathematics. 80 (2nd ed.). Springer-Verlag. p. 16. ISBN 0-387-94461-3.
- ↑ Lyndon, Roger C.; Schupp, Paul E. (2001). Combinatorial group theory. Classics in Mathematics. Springer-Verlag, Berlin. p. 87. ISBN 3-540-41158-5. https://books.google.com/books?id=cOLrCAAAQBAJ.
 |  |
