  We revisit, in an elementary way, the classical statement of various ``Main
Conjectures'' for $p$-class groups $\mathcal{H}_K$ and $p$-ramified torsion
groups $\mathcal{T}_K$ of abelian fields $K$, in the non semi-simple case $p
\mid [K : \mathbb{Q}]$. The classical ``algebraic'' definition of the $p$-adic
isotopic components, $\mathcal{H}^{\rm alg}{K,\varphi}$, used in the
literature, is inappropriate with respect to analytical formulas. For that
reason we have introduced, in the 1970's, an ``arithmetic'' definition,
$\mathcal{H}^{\rm ar}{K,\varphi}$, in perfect correspondence with all
analytical formulas and giving a natural ``Main Conjecture'', still unproved
for real fields in the non semi-simple case. The two notions coincide for
relative class groups $\mathcal{H}_K^-$ and groups $\mathcal{T}_K$ since, in
$p$-extensions, transfer maps are injective for these groups but not
necessarily for real class groups. Numerical evidence of the gap between the
two notions is given (Examples A.2.2, A.2.3) and PARI calculations corroborate
that the true Real Main Conjecture for $K$ writes on the form $\#
\mathcal{H}^{\rm ar}_{K,\varphi} = \# (\mathcal{E}_K / \hat{\mathcal{E}}_K \,
\mathcal{F}_{\!K})_{\varphi}$, in terms of units $\mathcal{E}_K$,
$\hat{\mathcal{E}}_K$ (units of the strict subfields) and $\mathcal{F}_K$
(Leopoldt's cyclotomic units). A recent approach, conjecturing the capitulation
of $\mathcal{H}_K$ in some auxiliary cyclotomic extensions $K(\mu_\ell^{})$,
proves the difficult real case.
