Russian President Dmitry Medvedev on Sunday ordered an investigation into allegations of polling violations during December 4 parliamentary elections, his Facebook page said on Sunday, a day after crowds protested the vote in several major cities.

Tens of thousands of people in Moscow and around Russia demanded a rerun of the election on Saturday in the biggest opposition protests Russia has seen in years.

The opposition says the vote was rigged in favour of the ruling United Russia party, which won a slim majority in the State Duma Lower House.

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 popular social media site.

Protesters waved banners with slogans including The rats should go! and Swindlers and thieves -- give us our election back! during the Saturday rallies -- a clear statement of intent against United Russia and Prime Minister Vladimir Putin.

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, Medvedev added in his first comment since the rally.

Putin is bidding to return to the Presidency in March for a possible 12 more years, having already served two terms between 2000 and 2008.

Medvedev, who has been President since 2008, is an avid user of social networking sites Facebook and Twitter.

(Reporting By John Bowker; Editing by Melissa Akin)