Can anyone give an example where (asymmetric) crypto can go wrong due to selection of wrong groups? Basically the title says it all. It would be great if someone could tell give an example using provable security. More information about groups can be found at:

https://en.wikipedia.org/wiki/Group_(mathematics)

https://en.wikipedia.org/wiki/Group-based_cryptography
Basically, every time you choose a group where the required hard problem is not hard, then you will run into a problem. Even if we have a problem instance that is of size that is considered secure in the setting of asymmetric cryptography. 

Lets for instance implement a discrete logarithm style cryptosystem in the group $Z_n$ with addition and let $g$ be a generator. Then the discrete logarithm problem in this group is given $g$, $y$ you have to find the discrete logarithm $x=\log_g(y)$, i.e., compute $x$ such that $y\equiv xg \pmod n$. However, you can efficiently solve this linear congruence by using Euclid. 

Consequently, the discrete logarithm problem in the group $(Z_n,+)$ is easy and thus every discrete logarithm based cryptosystem will be insecure in this setting.