Is Nancy Reagan showing the French a thing or two about where a woman's place is? An American woman's place? Better still, an American first lady's place?

Mrs. Reagan knows where her place is--with her man--even though only President Reagan initially was invited by the French to the seven-nation economic summit conference hosted by President Franc,ois Mitterrand opening Friday at Versailles. When they got the word that Mrs. Reagan was going to Paris anyway, the French subsequently amended the invitation to include her.

"No wives were invited because wives never come to summits," says one highly placed French official, "but since Reagan had his wife coming, we invited the others to bring their spouses."

Except for Hannelore Schmidt, wife of West German Chancellor Helmut Schmidt, no others have accepted, according to this same source.

"That means there will be no Mr. Thatcher (England) and no Mrs. Suzuki (Japan)," says the official, adding that there is no Mrs. Giovanni Spadolini (Italy's prime minister is a bachelor) and no Mrs. Pierre Trudeau "at the moment."

Even with an invitation, Nancy Reagan and "Loki" Schmidt will be able to spend only one night with their husbands at Versailles. The Reagans arrive in Paris tomorrow night and will entertain the Mitterrands at an American Embassy dinner Thursday. But starting Friday when the president moves to Versailles, the French have booked Reagan, Schmidt and the other leaders into more than 30 hours of official togetherness. They have left no room for spouses until Sunday night when Mitterrand hosts a glittering summit windup dinner in the chateau's historic Hall of Mirrors.

Mrs. Reagan will go to Normandy for D-Day observances earlier in the day and that night, after moving out of the American Embassy residence in the heart of Paris, will join her husband. It will be her first glimpse of his apartment at the Grand Trianon, the pink marble palace at the other end of the Grand Canal in the gardens of Versailles. Since the late 1960s the French government has been putting up important state visitors at the Trianon, which started out as Louis XIV and Madame Franc,oise de Montespan's favorite retreat from chateau court life.

For all their economic woes, the French have spared no expense in keeping their visitors comfortable, informed and entertained. They have renovated the 70-room Trianon's apartments, restored some historic works of art, emptied the Orangerie of its potted trees in order to turn it into a gigantic press center and installed one of the most up-to-date communications systems summiteers have ever seen. One gadget that fascinated recent visitors inspecting the conference room where Reagan and others will assemble is an electronic device that pictorially transmits handwritten messages to aides poised in readiness in nearby holding rooms.

Mitterrand's Sunday-night finale, with its mixture of pomp, pageantry and very important people, is definitely a status invitation this week in Paris, even among the megabuck crowd that sought to disassociate itself from his socialist government a year ago. Besides a spectacular fireworks display, Mitterrand's guests will be treated to a performance of the 17th-century opera, "Les Arts Florissants," by Marc-Antoine Charpentier in Versailles' jewel-like royal opera house. And that doesn't happen every night.