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Normal closure (group theory)

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

Properties and description

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Formally, if is a group and is a subset of the normal closure of is the intersection of all normal subgroups of containing :[1]

The normal closure is the smallest normal subgroup of containing [1] in the sense that is a subset of every normal subgroup of that contains

The subgroup is generated by the set of all conjugates of elements of in

Therefore one can also write

Any normal subgroup is equal to its normal closure. The conjugate closure of the empty set is the trivial subgroup.[2]

A variety of other notations are used for the normal closure in the literature, including and

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 [3]

Group presentations

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For a group given by a presentation with generators and defining relators the presentation notation means that is the quotient group where is a free group on [4]

References

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  1. ^ 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.
  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.