It seems that from [this webpage](http://www.map.mpim-bonn.mpg.de/Spin_bordism#Classification), the spin cobordism is equivalent to KO theory in low dimension.

If we denote the $p$-torsion part (mean $\mathbb{Z}_{p^n}$ for some $n$) $$\Omega_d(BG)_p.$$ 

**Question 1**:
Then do we have
$$\Omega_d^{spin}(BG)_p = ko_d(BG)_p?$$ for $p=2$ and free part, for $d\le 7$.

And
$$
\Omega_d^{spin}(BG)_p = \Omega_d^{SO}(BG)?
$$
for $p \neq 2$ and $p$ is an odd prime.

Namely, the 2-torsion and free part of $Mspin$ and $KO$ is the same. If there is an odd $p$ torsion, we need to consider localization at odd prime by $MSO$ cohomology. Is this correct?

**Question 2**:
If this is a statement about the spectra, not just about stable homotopy groups, and thus within these spin cobordism and ko theory, do they completely coincide for any dimensions $d$, instead of just $d \leq 7$?