When the senior U.S. general in Iraq announced the deaths of Saddam Hussein's sons Uday and Qusay in a shootout with U.S. forces on Tuesday, he said it should send a message to the Iraqi people that the Saddam regime is gone forever. The coalition hopes that message will help its effort to restore order by removing some of the inspiration behind the groups that have been attacking U.S. soldiers in recent weeks.

On a busy market street in downtown Baghdad, a woman who gives her name only as Friya is reluctant to answer a reporter's questions.

She says she is afraid to talk openly, because maybe Saddam Hussein will come back and execute her. She says he executed her son.

Friya is not willing to discuss politics or current events. But she will talk about her son, with grief that has not been softened by years. In a quiet voice, with frequent glances over her shoulder, Friya says her son was an ambitious young man of 26, who wanted to leave the country to finish his studies. The government tried to force him to become a member of the ruling Baath party, and he refused.

She says he left home one morning and never came back. She says the family asked about him, but he had just disappeared.

They found out later that he had been executed.

As heart-rending as her story is, Friya is far from alone in her grief. The United Nations and independent human rights groups say about 300,000 people were executed in the 24 years of Saddam's rule. That is more than 10,000 executions a year, on average.

Friya explains why the atrocities of the former regime make her afraid to speak her mind freely, even after the fall of Saddam Hussein.

"I do believe that because Saddam Hussein wasn't killed yet," she said. "Nobody killed or arrested him."

Friya's fear of Saddam's return is shared by many Iraqis. They spent 24 years in a police state with the Baath-party looking over their shoulders, and they remember the consequences of failure to toe the party line. Many believe that Saddam's loyalists are still among them, keeping track of what they say and do.

Journalists working in Iraq have found that many people will say one thing when interviewed in public, on the street, but something completely different when interviewed in the privacy of their own homes. Indeed, the VOA correspondent in Mosul on Wednesday reported Iraqis were chanting their support for Saddam in front of news cameras, but privately some said they were glad to see his two sons are dead.

Rumors circulate regularly in Baghdad, saying Saddam will return on a specific date. When that day passes, new rumors begin. Everyone has heard stories about how Saddam has been seen eating hummus in one town or drinking tea in another one.

The coalition's chief civilian administrator in Iraq, Paul Bremer, acknowledges that having Saddam Hussein on the loose is making it harder for the coalition to convince the Iraqi people that his reign is really finished.

"Having that issue unresolved gives these diehard remnants the opportunity to say to other people, Saddam is still alive and he's going to come back and so forth," said Mr. Bremer. "Well, he's not going to come back. He may be alive, but he's not going to come back."

On Tuesday, the senior U.S. general in Iraq, Ricardo Sanchez, said the killing of Saddam Hussein's sons should help convince Iraqis that Saddam and his regime are not coming back.

"I believe very firmly that this will in fact have an effect," said General Sanchez. "This will prove to the Iraqi people that at least these two members of the regime will not be coming back into power, which is what we have stated over and over again. And we remain totally committed to the Hussein regime never being in power and tormenting the people."

The man who told the coalition where to find Saddam's sons will reportedly receive the $30 million reward the United States has offered, 15-million dollars for each. U.S. officials hope the $25 million reward offered for Saddam Hussein himself will help convince someone to expose his hiding place before long.

Senior coalition officials blame Saddam loyalists for the daily attacks on U.S. troops, and for much of the sabotage of Baghdad's electricity grid and water system. And recently, Iraqis who cooperate with the coalition forces, such as policemen and interim mayors, have been targeted for assassination.

Coalition authorities say the continuing intimidation and the popular belief that Saddam may someday return are slowing down their recovery efforts and making it hard for them to win the trust of the Iraqi people.

But some analysts believe it would be a mistake to blame all the troubles in Iraq on remnants of the former regime. They say other elements, from criminal gangs to religious extremists could be exploiting the country's general instability to make their own plays for power or influence.

Political scientist Ehab Samir Bajis says the best thing the U.S. troops can do is prove by their actions that things have changed.

"The Americans must act cleverly now, because they must remove the idea from the Iraqi people about the former regime," he said. "They can give some beautiful example for the Iraqis and the Iraqis will be very friendly, really, if they see some difference between the former regime's acts and the Americans."

The U.S. military calls that "winning the hearts and minds" of the Iraqi people. And even if the military is somewhat successful in that effort, many Iraqis want the foreign forces out of Iraq as soon as possible anyway. U.S. officials say the troops can not leave as long as Saddam loyalists continue to make trouble, something they hope will subside as the coalition captures or kills more senior members of the old regime.