Why is Multiplicative group used in RSA or Euler's theorem but not additive? I am on the verge to understanding RSA, but suddenly a question popped into mind. When we are calculating $U(N)$ i.e $U(PQ)$, we are taking invertible elements that are co-prime to $N$. For example, $U(5\times2)$ becomes $\{1,3,7,9\}$. We are ultimately never using the inverses in $U(N)$ in either key generation or encryption. But just the order of $U(N)$ is what we need. We are using invertible elements in the order of $U(N)$ as a part of key generation, And that part I understand why, but the never using the invertible elements within $U(N)$ itself. So thereby the question, Why we don't use additive groups? Is it a security thing? 
PS, I don't know much about additive groups, just the multiplicative group as it's a part of RSA and I am studying RSA. Thanks :)
Why we don't use additive groups? Is it a security thing?


Yes, it's a security consideration. If we used the additive group $(\Bbb Z_N,+)$ rather than $(\Bbb Z_N^*,*)$  for RSA, public encryption would go $M\mapsto C=e\,M\bmod N$ rather than $M\mapsto C=M^e\bmod N$. Problem is, decryption would be trivial since anyone with the public key $(N,e)$ could compute $d=e^{-1}\bmod N$ (e.g. using the extended Euclidean algorithm) and then decrypt as $S\mapsto M=d\,S\bmod N$, without knowing the factorization of $N$.

This is related to the order of $(\Bbb Z_N,+)$ being $N$ part of the public key $(N,e)$, when the order of $(\Bbb Z_N^*,*)$ is $\Phi(N)$, which is not easily obtained from the public key.