Arthur Jones, a Holocaust denier described as a Nazi by the Illinois Republican Party, is poised to win the Republican primary in the state’s Third Congressional District. He is running unopposed, all but guaranteeing he will soon be the official Republican candidate for the House of Representatives. Mr. Jones is a longtime fixture of the white supremacist movement in the Midwest and a former member of the American Nazi Party. He was a far-right student activist at the University of Wisconsin, where he said he edited a conservative newspaper and attended meetings of the Young Republicans and a National Socialist group. In 1976, he ran for mayor of Milwaukee as a member of the National Socialist White People’s Party. He has previously run for the Republican nomination in the Third District of Illinois, in 1998, 2006, 2008, 2012 and 2016. But he has never won.