  In this paper, we study completely faithful torsion
$\mathbb{Z}_p[[G]]$-modules with applications to the study of Selmer groups.
Namely, if $G$ is a nonabelian group belonging to certain classes of polycyclic
pro-$p$ group, we establish the abundance of faithful torsion
$\mathbb{Z}_p[[G]]$-modules, i.e., non-trivial torsion modules whose global
annihilator ideal is zero. We then show that such $\mathbb{Z}_p[[G]]$-modules
occur naturally in arithmetic, namely in the form of Selmer groups of elliptic
curves and Selmer groups of Hida deformations. It is interesting to note that
faithful Selmer groups of Hida deformations do not seem to appear in literature
before. We will also show that faithful Selmer groups have various arithmetic
properties. Namely, we show that faithfulness is an isogeny invariant, and we
prove "control theorem" results on the faithfulness of Selmer groups over a
general admissible $p$-adic Lie extension.
