"Brotherhood actions caused the deaths of three citizens when they clashed with residents," the interior ministry said in a statement on the nationwide clashes.
Riot police fired tear gas at stone-throwing protesters in several cities, swiftly clamping down on rallies after the midday Friday prayers, the usual time for Islamist demonstrations.
The military-installed government has banned protests by Brotherhood members demanding the reinstatement of deposed president Mohamed Morsi, after listing the movement as a terrorist organization on Wednesday.
One person was shot dead in clashes in the city of Samalut in Minya province, south of Cairo, a hospital administrator said.
Interior ministry spokesman Hany Abdel Latif told AFP police were investigating the man's death, adding that an officer was also wounded in clashes with Islamists in the province.
He accused the protesters of using firearms and petrol bombs.
Smoke rose from Al-Azhar university's student dormitory in Cairo as police fired tear gas against protesters pelting them with rocks from inside the building.
A security official said protesters torched several police cars in Cairo and in Minya province.
Police seemed intent on hunting down any suspected protesters on Friday, with more than a dozen armoured vehicles racing to a Cairo mosque after they received reports of Brotherhood members gathering there.
Police also clashed with protesters in the Suez Canal city of Ismailiya, an AFP correspondent said.
The interior ministry overnight said a man was killed in clashes around Al-Azhar University between Islamist students and civilians who oppose them.
The Brotherhood, which condemned Tuesday's suicide bombing, has denounced its listing as a terrorist organization and vowed to continue staging peaceful rallies.
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