PARIS - French President Francois Hollande's Socialists won an absolute parliamentary majority on Sunday, strengthening his hand as he presses Germany to support debt-laden euro zone states hit by austerity cuts and ailing banks.The Socialist bloc secured between 296 and 320 seats in the parliamentary election runoff, according to reliable projections from a partial vote count, comfortably more than the 289 needed for a majority in the 577-seat National Assembly.Party unity over Europe is keyHollande was elected on May 6 after voters punished Sarkozy for failing to bring down rampant unemployment during a three-year economic crisis and for a personal manner many dislike.The parliamentary win, coming despite low turnout of 46.2 percent, leaves Hollande free of having to rely on opposition conservatives or eurosceptic hard leftists and should also leave his largely social democratic and pro-Europe cabinet intact.Hollande's Socialist lawmakers can also count on the backing of deputies from the Greens Party, which is allied with them.Hollande needs every lawmaker's vote he can get if a public finance audit due by end-June shows that France must slow its spending promises to meet its deficit goals.But one key supporter, his former companion Segolene Royal, who ran unsuccessfully for the presidency in 2007, was beaten on Sunday by a left-wing rebel backed by Hollande's current partner in the western city of La Rochelle. Royal's defeat could create an awkward problem for Hollande, who will need to find a face-saving role for her.The president also needs to keep Eurosceptical Socialist lawmakers behind him if he agrees to Germany's demand for deeper political integration in the euro zone. Prime Minister Jean-Marc Ayrault said on Sunday any steps towards federalism would need to ensure that national parliaments provided oversight.Hollande is still short of the two-thirds majority he would need for constitutional changes that would be required for example to give EU institutions more power over the budget."Hollande's biggest political test will be to keep his party united if he is forced to adopt economic policies that are unpopular with the electorate," political analyst Antonio Barroso of Eurasia Group said in a note to clients.The result means Hollande won't need to rely on the environmentalist Greens, projected to win 20 seats, or the Communist-dominated Left Front, likely to have just 10 deputies, to pass laws. The center-left already controls the upper house of parliament, the Senate.The far-right anti-immigration National Front achieved a breakthrough, winning its first parliamentary seats since the late-1980s. Its charismatic leader, Marine Le Pen, narrowly lost her race in a working-class northern town, but Marion Marechal Le Pen, 22, granddaughter of party founder Jean-Marie Le Pen, was elected in the southern town of Carpentras.The left-wing triumph gives the Socialists more power than they have ever held as Hollande pushes for new tools to stimulate growth in the sickly euro zone and a European banking union that would protect depositors and states if banks fail.Political turmoil in Greece, where parties that support the country's international bailout seemed set for a wafer-thin win on Sunday that would leave many problems unresolved, is piling pressure on Europe's leaders to act to contain the bloc's debt crisis at a summit later this month.Sunday's victory may help Hollande secure parliamentary backing for steps towards a euro zone fiscal union that Berlin is demanding as a condition for agreeing to his push for a growth pact and reforms to improve financial stability.The Socialist leader flies to Mexico on Monday for a Group of 20 summit that will be dominated by the euro zone's woes as a rift with the bloc's paymaster Germany over how to resolve the crisis has sparked a rare public squabble.Hollande, a pro-European social democrat, has broken with a Franco-German power duopoly established under his predecessor Nicolas Sarkozy and is siding with southern euro zone states, calling for more flexibility on deficit targets.He also wants a European banking union giving the European Central Bank power to supervise cross-border banks, with a joint deposit guarantee and a resolution fund, intended to ensure that bank collapses hit shareholders before taxpayers.With the right now severely weakened, Hollande's strong hand will be a boon as he readies a mass of legislation for the weeks ahead to raise taxes and ratify a European Union fiscal discipline pact.