  A comprehensive study of various ground-state properties of neutron-rich and
neutron-deficient Mg isotopes with $A$=20-36 is performed in the framework of
the self-consistent deformed Skyrme-Hartree-Fock plus BCS method. The
correlation between the skin thickness and the characteristics related with the
density dependence of the nuclear symmetry energy is investigated for this
isotopic chain following the theoretical approach based on the coherent density
fluctuation model and using the Brueckner energy-density functional. The
results of the calculations show that the behavior of the nuclear charge radii
and the nuclear symmetry energy in the Mg isotopic chain is closely related to
the nuclear deformation. We also study, within our theoretical scheme, the
emergence of an "island of inversion" at neutron-rich $^{32}$Mg nucleus, that
was recently proposed from the analyses of spectroscopic measurements of
$^{32}$Mg low-lying energy spectrum and the charge rms radii of all magnesium
isotopes in the $sd$ shell.
