  The dark matter halo surface density, given by the product of the dark matter
core radius ($r_c$) and core density ($\rho_c$) has been shown to be a constant
for a wide range of isolated galaxy systems. Here, we carry out a test of this
{\em ansatz} using a sample of 17 relaxed galaxy groups observed using Chandra
and XMM-Newton, as an extension of our previous analysis with galaxy clusters.
We find that $\rho_c \propto r_c^{-1.35^{+0.16}_{-0.17}}$, with an intrinsic
scatter of about 27.3%, which is about 1.5 times larger than that seen for
galaxy clusters. Our results thereby indicate that the surface density is
discrepant with respect to scale invariance by about 2$\sigma$, and its value
is about four times greater than that for galaxies. Therefore, the elevated
values of the halo surface density for groups and clusters indicate that the
surface density cannot be a universal constant for all dark matter dominated
systems. Furthermore, we also implement a test of the radial acceleration
relation for this group sample. We find that the residual scatter in the radial
acceleration relation is about 0.32 dex and a factor of three larger than that
obtained using galaxy clusters. The acceleration scale which we obtain is
in-between that seen for galaxies and clusters.
