normal closure

Let S be a subset of a group G. The normal closurePlanetmathPlanetmath of S in G is the intersection of all normal subgroupsMathworldPlanetmath of G that contain S, that is


The normal closure of S is the smallest normal subgroup of G that contains S, and so is also called the normal subgroup generated by S.

It is not difficult to show that the normal closure of S is the subgroupMathworldPlanetmathPlanetmath generated by all the conjugates of elements of S.

The normal closure of S in G is variously denoted by SG or SG or SG.

If H is a subgroup of G, and H is of finite index in its normal closure, then H is said to be nearly normal.

Title normal closure
Canonical name NormalClosure1
Date of creation 2013-03-22 14:41:50
Last modified on 2013-03-22 14:41:50
Owner yark (2760)
Last modified by yark (2760)
Numerical id 9
Author yark (2760)
Entry type Definition
Classification msc 20A05
Synonym normal subgroup generated by
Synonym conjugate closure
Synonym smallest normal subgroup containing
Related topic NormalizerMathworldPlanetmath
Related topic CoreOfASubgroup
Defines nearly normal