  Let k be a number field, p$\ge$2 a prime and S a set of tame or wild finite
places of k. We call K/k a totally S-ramified cyclic p-tower if Gal(K/k)=Z/p^NZ
and if S non-empty is totally ramified. Using analogues of Chevalley's formula
(Gras, Proc. Math. Sci. 127(1) (2017)),we give an elementary proof of a
stability theorem (Theorem 3.1 for generalized p-class groups X\_n of the
layers k\_n$\le$K:let $\lambda$=max(0, \#S-1-$\rho$) given in Definition 1.1;
then\#X\_n = \#X\_0 x p^{$\lambda$ n} for all n in [0,N], if and only if
\#X\_1=\#X\_0 x p^$\lambda$. This improves the case $\lambda$ = 0 of Fukuda
(1994), Li--Ouyang--Xu--Zhang (2020), Mizusawa--Yamamoto (2020),whose
techniques are based on Iwasawa's theory or Galois theory of pro-p-groups. We
deduce capitulation properties of X\_0 in the tower (e.g. Conjecture 4.1).
Finally we apply our principles to the torsion groups T\_n of abelian
p-ramification theory. Numerical examples are given.
