Let $G$ be a a finite group and $S$ a subnormal subgroup of $G$. The lenght of a fastest chain of subgroups $(U\_i)\_{1\le i\le n}$ such that $U\_1=S$, $U\_i$ normal in $U\_{i+1}$ and $U\_n=G$ is called the defect of subnormality of $S$. Examples in $S\_4$ Show that the series of repeated normalizers can be stable and is not reaching the whole group for a subnormal subgroup. But for nilpotent groups this series reaches $G$. My question: Is the length of this series exactly the defect of subnormality for nilpotent groups?



Defect of subnormality and repeated normalizer series
Let H be a cyclic group of order 2 and let K be the dihedral group of order 8 with y a non-central element of order 2. Then let G be the wreath product of H by K with base group B.
Then $J = \langle H,Hy\rangle$ is normal in B which is normal in G. But the normalizer of J in G is $B\cdot \langle y \rangle$, which is not normal in G.


