In the paper by Nakamoto, miner can accept the mined block only if all transaction are valid.
Assume that there are 5 miners in the world. 4 of 5 are honest that have 10% of 100% hash power and 1 of 5 is malicious that has 90% of all hash power.
Let's say group A consists of honest nodes (4 of 5). Let's say group B consists of malicious node (1 of 5).
Group B wants to create a block which has a fraud transaction (e.g. Alice wants to spend 1000 BTC but she actually does not have it). Group B is successively creating and propagating some blocks with huge hash power, but Group A does not accept these blocks because transactions in blocks are invalid. Yes, 51% attack is not accepted with only hash power. So, length of chain of group A is short but group B is long. But group A does not accept chain of group B.
So my conclusion is that 51% attack makes a mess in the network, not coin control.
I'm confused. Please help me understand where I am wrong.