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In addition to his SOSP donation, Mr. Soros in 2006 also supported the project’s candidates in Ohio, Jennifer Brunner, to whom he gave $2,500, and in Minnesota, Mark Ritchie, who got $250. Both won.

In 2006, SOSP helped elect Democratic secretaries of state in Ohio, Minnesota, New Mexico, Nevada and Iowa while its candidates lost in Colorado and Michigan. In 2008, the group backed winning candidates in Montana, West Virginia, Missouri and Oregon. SOSP raised $280,316 and spent $278,224 in that two-year election period. It could not be determined how much the project raised additionally in donations for the candidate’s individual campaign funds.

In 2010, just two of the group’s seven candidates won in a Republican year — in Minnesota and California. It lost in Ohio, Colorado, Iowa, South Dakota and Michigan. The group said it raised $193,767 and spent $243,112. It could not be determined how much it raised in additional donations for individual candidates.

Minnesota is the prime example of the project’s success. Helping to elect Mr. Ritchie in 2006 and 2010, Democrats had one of their own making key decisions when the extremely close U.S. Senate race between incumbent Norm Coleman, a Republican, and his challenger, former comedian Al Franken, went to a recount in 2008.

Mr. Ritchie headed the canvassing board that conducted the recount. Mr. Coleman initially had a lead of 206 votes out of 2.9 million cast, but after the recount, the board decided Mr. Franken had won by 225 votes. Republicans criticized Mr. Ritchie and the canvassing board, but the Minnesota Supreme Court unanimously upheld the finding.

Republican Mary Kiffmeyer, who lost to Mr. Ritchie in 2006, said SOSP’s involvement contributed to her defeat.

“They absolutely had an effect,” said Ms. Kiffmeyer, now a GOP state representative. She said she was leading by 17 percentage points the week before the election, when SOSP and its allies spent hundreds of thousands of dollars on targeted television ads and mailings. She said she had no time or money to respond to the last-minute attack ads, which linked her to Mr. Bush.

Ms. Kiffmeyer said she was limited to raising no more than $500 from an individual and spending just $250,000, but the SOSP had no such limits.

Mr. Soros, who lives in New York, did not donate directly to SOSP in 2006, but he was a serious donor to other important groups in Minnesota during the 2006 campaign. He gave $200,000 to America Votes-Minnesota, which led a get-out-the-vote drive just before the election — more than half of what it raised in 2006. He also gave $10,000 to the Minnesota Democratic-Farmer-Labor Party on whose ticket Mr. Ritchie ran.

“I want to thank the Secretary of State Project and its thousands of grass-roots donors for helping to push my campaign over the top,” Mr. Ritchie wrote. “Your wonderful support — both directly to my campaign and through generous expenditures by the strategic fund — helped me get our election reform message to Minnesota voters. And the voters overwhelming cast their ballot to protect our democracy on election day.”