AROUND 8 p.m. on Sept. 4, as night fell over the southern Japanese island of Okinawa, a 12-year-old girl was raped by several men. This is not the sort of crime that often happens in Japan, one of the most crime-free societies in the world, and many Japanese are aghast. The accusation that three American servicemen were responsible is making the incident an international issue. In recent days American officials in Japan have been quick to apologize for ''the suffering this crime has brought to the child, her family, and the people of Okinawa Prefecture,'' in the words of one American Embassy statement. The incident has given opponents of the American military presence in Japan new cause to complain, even though the number of crimes attributed to US personnel has been declining in recent years. But the complex security relationship between the United States and Japan is a sensitive subject these days, and it appears that officials of both countries do not want the furor over this incident to escalate. Next door, in South Korea, public furor over alleged misconduct by American troops this summer has prompted the South Korean government to ask for a review of the agreement that governs the presence of the 37,000 American troops in that country. One American official in Tokyo, speaking on condition of anonymity, says the United States does not want to see a similar request made by the Japanese. So far, the Japanese government has said it is comfortable with the so-called Status of Forces Agreement (SOFA). Nonetheless, US Ambassador to Japan Walter Mondale and Japanese Foreign Minister Yohei Kono yesterday announced that a bilateral panel of experts would be formed to examine how the US-Japan SOFA could be implemented more effectively. In both South Korea and Japan, the thorniest aspect of the SOFAs concerns the custody of American military personnel who are accused of crimes. The South Korea SOFA allows the US government to request custody of American personnel at any time, even after a conviction. The Japan version is more accommodating of Japanese sovereignty, allowing local law-enforcement agencies to claim custody of an American suspect as soon as an indictment has been filed. Many Japanese are outraged that the three servicemen implicated - but not yet indicted - in the Okinawa rape are still in US custody. In a resolution earlier this month, the legislative assembly in Naha, the Okinawa prefectural capital, said it would not ''tolerate the US military authorities' refusal to hand over the suspects in this case on the basis of the SOFA.'' American officials have said they are allowing Japanese authorities to question the men at any time, and will hand them over if indictments are filed. Some Japanese have been surprised at the speed and tone of the American response. Officials in Washington, Tokyo, and Naha have apologized, and the commander of US forces in Japan, Lt. Gen. Richard B. Myers, said Tuesday that ''this terrible tragedy was an outrageous attack [on] humanity and makes all of us wearing the US military uniform deeply ashamed.'' Aside from occasionally referring to the servicemen as ''suspects'' and the crime as an ''alleged rape,'' there have been remarkably few assertions of the innocence of the men involved. An Okinawa prefectural police official said in a telephone interview yesterday that two of the servicemen had confessed to US investigators. ''I'm surprised at such a quick apology from the US government,'' says Seigen Nagayoshi, a lawyer in Naha and the secretary-general of the Okinawa Human Rights Association, a group that has campaigned for the withdrawal of the 29,000 US troops stationed there. He says that the US wants to mollify Okinawans in order to maintain the troops' presence there. The US is reviewing its military role in Japan. Some academics and politicians in the US have called for a significant downsizing of the approximately 45,000 troops that the US maintains in Japan, arguing that the arrangement is a relic of the cold war and that the Japanese should be encouraged to provide for their own defense. The forces stationed on the Japanese archipelago are the most important element in the American security presence in the Pacific. A policy paper released earlier this year by the Pentagon said the US should keep the 100,000 troops stationed in the region. Top American and Japanese officials have repeatedly affirmed their commitments to maintain the arrangement. In recent years, the US has urged Japan to contribute increasing amounts of money to cover the costs of the troops. In the 1995 fiscal year Japan is contributing $4.8 billion, which American officials say covers 70 percent of US military costs. One reason for the rapid American reaction to the Okinawa incident may be a set of upcoming meetings that will include discussions of the security relationship. Next week top US and Japanese Cabinet ministers gather in New York. President Clinton is coming to Japan in November to meet his Japanese counterpart.