While reading about the Burnside problem, I thought of the following question:



```
 If every proper subgroup of G is finite, does it follow that G is also finite?

```

Despite extensive searching (and thinking), I am unable to find a solution. (I suspect that the answer is no)



An infinite group such that every proper subgroup is finite?
No. The direct limit of the cyclic groups of order $p^n$ is infinite, but every proper subgroup of it is finite.


