By Jeremy Bowen
BBC Middle East editor
For the last 12 months, the suggestion that they should try to end their argument with Hamas has been guaranteed to get a testy response from senior figures close to Palestinian Authority President Mahmoud Abbas.
The humiliation of June 2007 will not easily be forgotten by Fatah's people
One of his senior ministers exploded with such fury whenever I asked him about it that his voice sent the dials on the BBC's recording equipment hurtling into the red.
They all felt the humiliation inflicted by Hamas when it seized power in Gaza from Fatah exactly a year ago as if it was a personal physical pain.
Hamas men with guns roamed through Fatah buildings, shooting up the offices of men who had been grandees.
The fighter who emptied his Kalashnikov into the desk of Mohammed Dahlan, until that day the Fatah strongman in Gaza yelled "This is the fate of traitors like the scumbag Dahlan" as he pulled the trigger, and it was recorded and put on television for all to see.
Divide and rule?
For a year now American, Israeli and British officials have had an easy answer when they have been asked whether a dialogue with Hamas should be considered, since the policy of isolating them was causing such human misery in Gaza.
They would lean forward, and point out that there was no stronger supporter of the boycott of Hamas than Mr Abbas, who is also the leader of Fatah, the faction that dominated Palestinian politics from the 1960s until the rise to power of Hamas.
The split between the two main Palestinian factions has been a source of strength, short term at any rate, for Israel
Mr Abbas laid down a strict precondition for dialogue. He said that it was impossible until Hamas revoked what Fatah people always referred to as the "illegal coup" in Gaza.
For their part Hamas leaders said they wanted to talk to Fatah, in the interests of national unity, but would not hear of preconditions.
The split between the two main Palestinian factions has been a source of strength, short term at any rate, for Israel.
Some Israelis point out that longer term it might not be. How do you make peace with a divided people?
Signs of change
Since the end of last year, Israel and the Palestinians led by Mr Abbas have been involved in a new peace process, sponsored by the Americans and inaugurated by President Bush with quite some ceremony at Annapolis in Maryland.
But Mr Abbas has declared his desire for peace with Israel for many years. The Palestinians who are at war with Israel belong to Hamas and their allies.
Mr Abbas will need Arab support if dialogue with Hamas is to work
So far none of this has changed. But there are signs that it might.
Mr Abbas seems to have dropped his demand that Hamas give up power in Gaza as a precondition for talks.
Last week he called for dialogue and then, over the weekend, there were contacts between the two sides in Senegal.
At the same time President Abbas went to Saudi Arabia to see King Abdullah and then to Cairo to see President Mubarak of Egypt. He will need their support if he does want to make a deal with Hamas.
If it happens, it will not go down well with the Americans and the Israelis and many of their allies. As far as they are concerned, Mr Abbas has been very useful, lining up with them against Palestinians that they regard as terrorists.
American policy has been built around building Mr Abbas up at the expense of Hamas. It is not clear whether Washington has a Plan B.
Mr Abbas seems to be responding to a desire for unity among Palestinians.
Some of his senior officials say he's fed up with Israel's continued expansion of Jewish settlements, and the lack of progress in the Annapolis peace talks.
But there are signs that talking to Hamas is controversial within Fatah as well.
One Palestinian negotiator said nothing had changed. Hamas, he said, still had to revoke its coup in Gaza before there could be unity.
The United States might not need to find a Plan B just yet, but the men who want to be the next US president might want to start thinking of one, just in case.