If they start not connected to the main group they can not make transactions for a 2 simple reasons, there are no transactions for them to process because no one has any assets to send, and if anyone does have any assets to send, the initial set of allowed counter-signers isn't present because they are on the main group.
How can the smaller group know about the main group until it rejoins it?
I maintain that unless you have a fixed set of ledger producing nodes, or a deterministic selection policy (which has its own set of problems), you must be able to deal with forks, because they will be inevitable.
A block is 1 persons vote, the miner who made it. There can not be a majority in a set of 1, and I can come and change that "vote" at any time providing I produce a block(s) with a greater amount of work done.
1 block represents 25 BTC of hashing power, not one person. In addition, subsequent blocks placed on top of that block are *additional votes* for that chain.
Are you sure? Who is presenting all these Byzantine failures that total 50%? It is proven by people many times smarter than you, I and everyone else here that the limit of true Byzantine tolerance in any trustless information system is 33%. Are you saying they are wrong? That Satoshi has achieved the equivalent of faster than light travel in information theory?
Byzantine failures are basically misinformation, through whatever means. The truth in POW is the longest chain of work, and since a majority is >50%, POW can withstand 50% byzantine failures. That is the true genius of satoshi.
With all due respect, I'm not going to throw away 2 years of work on the say so of someone I've never even spoken to before 3 days ago. This isn't theoretical work, as we've been actively testing this model for failures for 9 months or more now.
That is your prerogative, of course. I know it is very hard to look at your own work objectively, but it is also essential to do so, no matter the cost.