In the 1970s, after years of brutal repression by the state, the Egyptian president at the time, Anwar el-Sadat, permitted the Brotherhood to operate quietly and to open a Cairo office, and the Brotherhood formally renounced violence as a means of achieving power in Egypt. The group did not, however, reject violence in other circumstances, and its leaders have endorsed acts of terrorism against Israel and against American troops in Iraq.

A prominent Brotherhood thinker, Sayyid Qutb, who was imprisoned by the Egyptian government and executed in 1966, was an important theorist of violent jihad and a spiritual progenitor of Osama bin Laden, the founder of Al Qaeda, and Anwar al-Awlaki, the American-born radical preacher now hiding in Yemen. But the Brotherhood took a different direction after Mr. Qutb’s death, and Qaeda leaders came to hold the organization in contempt.

A milestone in the Brotherhood’s evolution in Egypt came in 1984, when its leaders decided to compete in parliamentary elections. Since then, it has been alternately tolerated and repressed in Egyptian politics, where most estimates of its actual support begin at 20 percent of the electorate.

“The paradox has been that the better the Brotherhood performs, the more repression it has attracted,” Ms. Wickham said. After it won 88 seats in Parliament in the 2005 elections, Mr. Mubarak’s government responded with a new crackdown.

In an interview just before the current wave of protests began in Egypt, Essam el-Erian, a leading figure in the Brotherhood, said the group did not seek to monopolize power. “We want an atmosphere for fair competition now that can allow us to compete for power in the future,” Mr. Erian said. “And we want stability and freedom for people, not chaos.”

The Brotherhood, whose leaders are mostly much older than the protest organizers, joined the demonstrations only after they were under way. The hesitancy may reflect in part the grim history of the state’s ruthlessness, said Abdel Halim Qandil, the general coordinator of Kifaya, a secular opposition movement.

“The Brotherhood was rebuilt over the last three decades as a social religious movement,” Mr. Qandil said. “They are having difficulty transforming that into a political movement.”