MOSCOW - President Dmitry Medvedev ordered an investigation on Sunday into allegations of fraud in Russia's parliamentary election, one day after tens of thousands of protesters demanded it be annulled and rerun.Medvedev responded on his Facebook site to the protesters' complaints that the December 4 election was slanted to favor of his and Prime Minister Vladimir Putin's United Russia party, but did not mention their calls for an end to Putin's rule.RELATED:Gorbachev calls for retake on Russian electionsAnalysis: Poor election shows Putin's vulnerability"I do not agree with any slogans or statements made at the rallies. Nevertheless, instructions have been given by me to check all information from polling stations regarding compliance with the legislation on elections," Medvedev said in a post on the social media site."Citizens of Russia have freedom of speech and freedom of assembly. People have a right to express the position that they did yesterday. It all took place within the framework of the law," he added.His statement was a sign that the Russian leadership feels under pressure after the biggest opposition protests since Putin rose to power in 1999. The protesters themselves used social media to organize their rallies.In a further sign of recognition that the people's mood has changed after years of tight political control by Putin, city authorities across Russia allowed Saturday's protests to go ahead and riot police hardly intervened.State television and other Russian channels also broadcast footage of a huge protest in Moscow, breaking a policy of showing almost no negative coverage of the authorities. But Medvedev had already indicated before the protests that he would call an inquiry, and a statement from Putin's spokesman, Dmitry Peskov, gave no indication that the prime minister was about to make big concessions to the protesters."We respect the point of view of the protesters, we are hearing what is being said, and we will continue to listen to them," Peskov said in a statement released late on Saturday.That is unlikely to appease protesters who issued a list of demands at the Moscow rally, which police said was attended by 25,000 people and the organizers said attracted up to 150,000.Protesters demanding not only voting investigation, but also reelections The demands included much more than just an investigation into the conduct of the election, which international monitors and the United States said was slanted to help United Russia secure a majority in the State Duma lower house.The protesters demanded a rerun of the election, the sacking of Central Election Commission chief Vladimir Churov and the release of people they define as political prisoners. The organisers also called for a new day of protests on Dec. 24. "I am happy. December 10, 2011 will go down in history as the day the country's civic virtue and civil society was revived. After 10 years of hibernation, Moscow and all Russia woke up," Boris Nemtsov, an opposition leader, wrote in his blog."The main reason why it was such a big success is that a feeling of self-esteem has awakened in us and we have all got so fed up with Putin's and Medvedev's lies, theft and cynicism that we cannot tolerate it any longer ... Together we will win!"It may not be that simple. The opposition has long been divided, most mainstream parties have little or no role in the rallies and keeping them up across the world's largest country is hard at the best of times but especially in winter.Most Russian political experts say Putin, the former KGB spy who has dominated the world's largest energy producer for 12 years, is in little immediate danger of being toppled, despite anger over widespread corruption and the gap between rich and poor.But they say the 59-year-old leader's authority has been damaged and may gradually wane after he returns as president in an election next March that he is still expected to win.Although opinion polls show he is Russia's most popular politician, the protests indicate how deep feelings are over the Dec. 4 election. The biggest were in Moscow and St Petersburg, the two biggest cities and the main centers of Russia's middle class, but smaller rallies took place across the country.Putin has tough task ahead"Putin has a formidable task. He has lost Moscow and St Petersburg, crucial cities where everything usually starts," said political analyst and author Liliya Shevtsova. "He looks out of touch."Putin, as president for eight years until 2008 and as prime minister since then, built up a strongman image by restoring order after the chaos in the decade after the Soviet Union's collapse in 1991. But he no longer seems invincible.He could release the state's purse strings to satisfy the financial demands of some critics but many of the protesters in Moscow are middle-class people demanding more fundamental changes, including relaxing the political system he controls.His charges last week that the United States encouraged the protesters and financed them provoked scorn on the Internet.Answering calls to protests on social media sites, a huge crowd gathered in Moscow's Bolotnaya Square on Saturday, many carrying white carnations as a symbol of protest. Some waved pictures of Putin and Medvedev saying: "Guys, it's time to go."Felix, 68, a retired military officer who declined to give his surname, said in Moscow he wanted Putin out, but had no hope this could be accomplished through elections. "There is no way to change those in power within the electoral system they have set up, so we need to use other methods," he said.