Sarah Huckabee Sanders wins the G.O.P. nomination for Arkansas governor.

It didn’t matter.

Mr. Kemp received by a large sufficient margin — he was main by practically 50 share factors when The Related Press referred to as the race — to keep away from a runoff. Mr. Perdue had anchored his candidacy on selling falsehoods concerning the final election, blaming Mr. Kemp each for Mr. Trump’s defeat and his personal loss in a 2021 runoff that gave Democrats management of the Senate. The result on Tuesday uncovered the bounds of Mr. Trump’s maintain on his celebration’s base.

Mr. Kemp’s victory units up a rematch of his 2018 battle with Stacey Abrams, who received the Democratic nomination unopposed on Tuesday, in what can be one of the crucial intently watched governor’s races within the nation this fall. His most pressing crucial is reuniting a Republican Celebration fractured by the divisive main.

The mass capturing at an elementary faculty in Texas that left 18 kids useless forged a shadow over the night, as Mr. Kemp delayed his victory speech till after President Biden addressed the tragedy.

In Georgia’s U.S. Senate race, Herschel Walker, the previous College of Georgia soccer star who was additionally recruited by Mr. Trump, received the Republican nomination and can face Senator Raphael Warnock, a Democrat, in November. The matchup within the fall, the result of which may tip management of the Senate, is a uncommon common election contest within the South pitting two Black candidates in opposition to each other.

Whereas Mr. Walker glided by means of the first, his tumultuous previous — together with accusations of home abuse and his exaggerated and false claims about his enterprise success — is predicted to obtain a extra thorough airing within the common election.

Each events count on Mr. Walker to be the topic of coming commercials questioning his competence and credentials. His Democratic opponent, Mr. Warnock, top-of-the-line fund-raisers within the nation, has been on the tv airwaves for months already, focusing largely on optimistic messages.

With practically 200 endorsements thus far, Mr. Trump has arrange the 2022 main season as a rolling referendum on his affect within the celebration. He has scored notable large successes, comparable to J.D. Vance in Ohio, and suffered defeats in Nebraska and Idaho. However no state thus far has been as a lot a spotlight for Mr. Trump as Georgia, the place he not solely got down to oust the governor but in addition Mr. Kemp’s allies throughout different statewide workplaces.

Within the secretary of state race, Brad Raffensperger, the Republican incumbent whom Mr. Trump pressured to “discover” the enough votes to overturn the election in early 2021, was forward of a Trump-backed challenger, Consultant Jody Hice. It was not clear, nevertheless, if he would clear the 50 p.c threshold wanted to avert a runoff.

The secretary of state serves as Georgia’s prime elections official, and the winner within the fall may have nice sway over how the 2024 presidential marketing campaign can be performed in a key battleground state.

Consultant Marjorie Taylor Greene of Georgia, the right-wing firebrand, handily forged apart a extra average challenger, carrying greater than two-thirds of the vote.

Whereas Georgia acquired prime billing on Tuesday, a number of different states held primaries.

In Texas, the final scion of the Bush political dynasty, Land Commissioner George P. Bush, was defeated within the state legal professional common’s race, dropping to the scandal-plagued incumbent, Ken Paxton. And in a Democratic contest alongside the border, Consultant Henry Cuellar, one of the crucial average Democrats within the Home, faces a progressive problem from Jessica Cisneros that has drawn nationwide consideration.

In Alabama, three Republican candidates have been vying to make the runoff to succeed retiring Senator Richard Shelby: Consultant Mo Brooks; Mike Durant, the helicopter pilot portrayed in “Black Hawk Down”; and Katie Britt, a former chief of employees to Mr. Shelby. Mr. Trump had initially endorsed Mr. Brooks early in 2021. However as Mr. Brooks sagged within the polls, he rescinded that endorsement.

Gov. Kay Ivey of Alabama was in search of to keep away from a runoff in opposition to two right-wing challengers.

And in Arkansas, Sarah Huckabee Sanders, the previous White Home press secretary and the daughter of former Gov. Mike Huckabee, has largely coasted by means of a Republican main for governor. Senator John Boozman of Arkansas is pushing to clear 50 p.c of the vote to keep away from a runoff in opposition to Jake Bequette, a former soccer star, in a race that has seen greater than $7 million in tv adverts.

In Georgia, Mr. Kemp had methodically locked up the assist of the state’s largest political gamers and donors in anticipation of a Trump-fueled main problem. He signed into regulation quite a few conservative priorities, together with a restrictive new voting regulation in 2021 and a gas-tax vacation that extends simply previous the first. He additionally expanded gun rights, raised trainer pay and dispatched tax rebate checks that went out in latest weeks.

All these maneuvers and extra left Mr. Perdue remoted, with little to lean on past the assist of Mr. Trump, who has seethed for 18 months about Mr. Kemp’s refusal to try to overturn the 2020 election ends in his state.

“His endorsement continues to be vital, but it surely comes with an asterisk,” mentioned Stephen Lawson, a Georgia-based Republican strategist, including that the Perdue marketing campaign’s failure was “proof optimistic your message must be greater than that.”

Even earlier than the election was over, Mr. Perdue’s personal allies started overtly criticizing the Perdue effort as lackluster. And plenty of nationwide Republicans rallied behind Mr. Kemp, because the Republican Governors Affiliation took the bizarre step of intervening to spend greater than $5 million on adverts within the main. Formidable potential 2024 Republican presidential candidates campaigned for him, together with former Gov. Chris Christie of New Jersey and, on the eve of the election, former Vice President Mike Pence.

Regardless of Mr. Trump’s ire and public assaults, Mr. Kemp by no means fired again on the former president. “I’ve by no means mentioned something dangerous about him,” Mr. Kemp mentioned on Monday. “I don’t plan on doing that. I’m not mad at him. I believe he’s simply mad at me.”

Now Mr. Kemp should hope that self-discipline pays off and that Mr. Trump — who mused at a rally in Georgia final fall that Ms. Abrams “is perhaps higher than having your present governor” — doesn’t actively work to torpedo him in November.

Mr. Walker tried to remain impartial within the governor’s main, declining to endorse and even say who he was voting for. The 60-year-old political newcomer averted any debates through the main and largely ignored his rivals in a subject that included Georgia’s agriculture secretary, Gary Black.

Mr. Black had warned repeatedly that the untested Mr. Walker would show unelectable — “he’ll by no means win,” Mr. Black mentioned in a latest interview — and that Republicans would come to remorse his anointment.

One of many methods Mr. Trump tried to ease the trail for Mr. Perdue was to push one other ally, Vernon Jones, out of the governor’s race and into an open congressional contest.

Georgia Republicans redrew district strains forward of 2022, cramming two Home Democrats into the identical suburban Atlanta district and diluting the Democratic power of one other seat within the state’s southwestern nook held by Consultant Sanford Bishop. Mr. Bishop, a Democrat, faces his first severe problem in years this fall.

Consequently, Republicans may develop their share of the congressional delegation from 8 to six to 10 to 4.

Within the main that positioned two Democratic incumbents in opposition to one another, Consultant Lucy McBath was going through Consultant Carolyn Bourdeaux on Tuesday.