Normal closure (group theory) explained

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 \operatorname{ncl} _{G(S)} of S is the intersection of all normal subgroups of G containing S :^[1] $$\backslash operatorname\_G(S)\; =\; \backslash bigcap\_\; N.$$

The normal closure

\operatorname{ncl} _{G(S)} is the smallest normal subgroup of G containing S, in the sense that \operatorname{ncl} _{G(S)} is a subset of every normal subgroup of G that contains S.

The subgroup

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

Therefore one can also write$$\backslash operatorname\_G(S)\; =\; \backslash .$$

\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 or, 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) / \operatorname{ncl} _{F(S)} (R), where F(S) is a free group on S. ^[4]

Notes and References

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