  In this paper, we study the Selmer groups of two congruent Galois
representations over an admissible $p$-adic Lie extension. We will show that
under appropriate congruence condition, if the dual Selmer group of one
satisfies the $\M_H(G)$-property, so will the other. In the event that the
$\M_H(G)$-property holds, and assuming certain further hypothesis on the
decomposition of primes in the $p$-adic Lie extension, we compare the ranks of
the $\pi$-free quotient of the two dual Selmer groups. We then apply our
results to compare the characteristic elements attached to the Selmer groups.
We also study the variation of the ranks of the $\pi$-free quotient of the dual
Selmer groups of specialization of a big Galois representation. We emphasis
that our results \textit{do not} assume the vanishing of the $\mu$-invariant.
