This is definitely not my expertise, but here is a stab at it.

A solution to Hilbert's 5th Problem states that given a topological group that is also a manifold there is a unique way to give it the structure of a Lie group.

On the other hand, there are exotic structures on Lie groups (including compact ones).

So take a Lie group $G$ that admits an exotic structure. Let $M$ denote the exotic version of $G$. Then $G$ and $M$ are homeomorphic but not diffeomorphic. However, if $M$ admitted a smooth group operation, then we contradict the uniqueness part of the solution to Hilbert's 5th Problem.

This is definitely not my expertise, but here is a stab at it.

A solution to Hilbert's 5th Problem states that given a topological group that is also a manifold there is a unique way to give it the structure of a Lie group.

On the other hand, there are exotic structures on Lie groups (including compact ones).

So take a Lie group $G$ that admits an exotic structure. Let $M$ denote the exotic version of $G$. Then $G$ and $M$ are homeomorphic but not diffeomorphic. However, if $M$ admitted a smooth group operation, then we contradict the uniqueness part of the solution to Hilbert's 5th Problem.

This is definitely not my expertise, but here is a stab at it.

A solution to Hilbert's 5th Problem states that given a topological group that is also a manifold there is a unique way to give it the structure of a Lie group.

On the other hand, there are exotic structures on Lie groups.

So take a Lie group $G$ that admits an exotic structure. Let $M$ denote the exotic version of $G$. Then $G$ and $M$ are homeomorphic but not diffeomorphic. However, if $M$ admitted a smooth group operation, then we contradict the uniqueness part of the solution to Hilbert's 5th Problem.

This is definitely not my expertise, but here is a stab at it.

A solution to Hilbert's 5th Problem states that given a topological group that is also a manifold there is a unique way to give it the structure of a Lie group.

On the other hand, there are exotic structures on Lie groups (including compact ones).

So take a Lie group $G$ that admits an exotic structure. Let $M$ denote the exotic version of $G$. Then $G$ and $M$ are homeomorphic but not diffeomorphic. However, if $M$ admitted a smooth group operation, then we contradict the uniqueness part of the solution to Hilbert's 5th Problem.