Candidates who question the 2020 election results won a handful of key primaries on Tuesday, underscoring the continued political potency of the myth of a stolen election in US politics.

The most significant victory was in Arizona, where Mark Finchem, who was endorsed by Donald Trump, easily won the GOP nomination for secretary of state, placing him one step closer to overseeing elections in a key battleground state.

Finchem, who has self-identified with the far-right Oath Keepers, vigorously fought to block certification of Joe Biden’s legitimate victory in Arizona and has sought to overturn it ever since.

He told reporters on Tuesday he received a subpoena from the Department of Justice, which is investigating the January 6 attack, about a month ago. He has also been subpoenaed by the congressional committee investigating the attack.

Finchem joins prominent election deniers in Michigan, Nevada and Pennsylvania who have earned the Republican nomination for positions in which they would wield considerable power over elections.

In the Arizona gubernatorial primary, Kari Lake, a Trump-backed former news anchor who has made election misinformation a centerpiece of her campaign, narrowly led rival Karrin Taylor Robson on Wednesday morning.

Even before she took the lead in ballot counting, Lake, who has already alleged fraud in the vote, claimed victory.

“There is no path to victory for my opponent and we won this race, period,” Lake said at her election night party. On Wednesday morning she led by just over 11,300 votes with 20% of the vote left to count.

Blake Masters, a Trump-backed US Senate candidate in Arizona who has questioned election results also easily won the GOP primary to take on the Democratic senator Mark Kelly.

Rusty Bowers, the speaker of the Arizona house who faced censure from his party and Trump’s fury after testifying in front of the January 6 committee, lost his primary for state senate to a Trump-backed challenger.

There were other signs of how election conspiracies continue to dominate Arizona politics. In Maricopa county, a Republican candidate for the board of supervisors urged voters to steal pens the county provided to fill out ballots, a nod to a baseless fraud claim promoted on Gatewaypundit, a far-right website.

In Michigan, Peter Meijer, one of 10 Republicans to support Trump’s impeachment, lost a primary battle against John Gibbs, who served in the Department of Housing and Urban Development in the Trump administration.

In a debate last month, Gibbs said there were “mathematically impossible anomalies” in the 2020 race, which is not true. Meijer blasted Democrats for boosting Gibbs’s campaign as part of a strategy to elevate more extreme candidates who might be easier to beat in November.

Michigan Republicans nominated Tudor Dixon, a conservative commentator, to take on Gretchen Whitmer for governor. Dixon has said the 2020 election was stolen in Michigan, where Trump lost by more than 150,000 votes, but has been vague about what exactly she says went wrong.

In Missouri, Eric Schmitt, who lead a coalition of attorneys general urging the US supreme court to overturn the 2020 election, won the Republican nomination for US Senate. Trump endorsed “Eric” in the race, declining to say whether he was backing Schmitt or another challenger, Eric Greitens.

In Washington state, two US House Republicans who voted to impeach Trump appeared to be doing fairly well as votes continued to be counted. With about half of the vote counted, Jaime Herrera Beutler and Dan Newhouse were both leading Trump-backed opponents.