G.O.P. primary victories in Nevada set the stage for Trump-centered battles in the fall.

Republican voters in Nevada on Tuesday elevated conservative candidates who have ardently embraced Donald J. Trump’s false claims of election fraud, turning a key swing state into a contest this fall between embattled Democrats and Republicans who insist President Biden stole the 2020 election.

The victories in the Nevada primaries for Mr. Trump capped a series of elections on Tuesday that saw one South Carolina Republican lawmaker who had crossed Mr. Trump go down in defeat, another survive her Trump-backed challenge and a Hispanic Republican grab a South Texas House seat vacated by a Democrat.

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