The conjugate closure of *S* is always a normal subgroup of *G*; in fact, it is the smallest (by inclusion) normal subgroup of *G* which contains <*S*>, the subgroup generated by the elements of *S*. We can compare this to the *normalizer* of *S*, which is the *largest* subgroup of *G* in which <*S*> is normal.

If *S* = {*a*} consists of a single element, then the conjugate closure is a normal subgroup generated by *a* and all elements of *G* which are conjugate to *a*. Therefore, if *G* is a simple group, *G* is generated by the conjugate closure of any non-identity element *a* of *G*.