See [Smooth manifold with non-trivial inertia group? (wrt homotopy spheres)](https://mathoverflow.net/questions/79991/smooth-manifold-with-non-trivial-inertia-group-wrt-homotopy-spheres) for the definition of $\Theta\_n$ and inertia subgroups.


I'm wondering what can be said about Lie groups. If $M^n$ is an n-dimensional manifold with Lie group structure and $\Sigma^n$ is a homotopy n-sphere, is there a Lie group structure on $M\#\Sigma$ that is in some sense compatible with the original structure? If this new group structure is isomorphic to the old structure, this implies Lie group isomorphism, correct?


What I would like to see is that there is a canonically induced Lie group structure on $M\#\Sigma$ and that this structure is isomorphic to that of $M$, and hence the inertia group for Lie groups is the full $\Theta^n$



The inertia subgroup of `$\Theta_n$` for Lie groups
There is no natural Lie group structure on a connected sum of a Lie group and an exotic sphere. Where would it possibly come from?


Also, the implication you are trying to derive is wrong. 
Note that $S^3$ is a Lie group and hence so is a product of several $S^3$s.
I'm not an expert on surgery theory but it's well-known that the [inertia group of a product of spheres is trivial](http://www.ams.org/mathscinet/search/publdoc.html?pg1=MR&s1=0275453) which means that the opposite of the statement you are after holds: the connected sum of a product of spheres with an exotic sphere is *never* orientably diffeomorphic to the original manifold.
 So for example
the inertia group of $S^3\times S^3\times S^3$ is trivial while $\Theta\_9$ has order 8.


