WASHINGTON -- The intensifying rivalry between former U.S. President Donald Trump and his once fiercely loyal vice president, Mike Pence, was put on stark display Tuesday as the two gave dueling speeches in Washington on the future of the Republican Party.

Trump, in his first return to Washington since Democrat Joe Biden ousted him from the White House, repeated the false election fraud claims that sparked the Jan. 6 Capitol insurrection, while Pence, in a separate address, implored the party to move on from Trump's defeat.

Federal and state election officials and Trump's own attorney general have said there is no credible evidence the 2020 election was tainted. The former president's allegations of fraud were also roundly rejected by courts, including by judges he appointed.

"It was a catastrophe, that election," Trump nonetheless declared to an audience of cheering supporters at the America First Agenda Summit, about a mile from the White House he once called home.

Hours earlier, addressing a student conservative group, Pence said, "Some people may choose to focus on the past, but elections are about the future."

The speeches highlighted the divisions within the party between Trump loyalists who still refuse to accept the results of the 2020 election and other Republicans who believe the party should instead focus on the future heading into this fall's midterm elections and beyond.

And they come as both men have been laying the groundwork for expected presidential runs. Trump, in particular, has been teasing his intentions and said Tuesday that he "may just have to do it again" as he addressed a group of former White House officials and Cabinet members who have been crafting an agenda for a possible second Trump administration.

Pence, once Trump's loyal vice president, spoke about his own "Freedom Agenda" as he presented a different vision for the party at a conference nearby.

"I believe conservatives must focus on the future to win back America. We can't afford to take our eyes off the road in front of us because what's at stake is the very survival of our way of life," he said in an address to Young America's Foundation, a student conservative group.

Trump, too, said America's survival was at stake. In a speech billed as focused on public safety, he painted a dark picture of a nation in decline and one in imminent danger from rising crime. Among his proposals, he called for executing drug dealers, sending the homeless to tent cities on the outskirts of cities, and expanding his Southern border wall.

Biden joined in -- on Twitter -- dismissing Trump's claim to have been a law-and-order president.

Referring to the Capitol riot, he tweeted: "I don't think inciting a mob that attacks a police officer is `respect for the law.' You can't be pro-insurrection and pro-cop -- or pro-democracy, or pro-American."

Trump, in his remarks, also spent plenty of time airing his usual grievances even as some advisers have urged him to move on.

"If I renounced my beliefs, if I agreed to stay silent, if I stayed home and just took it easy, the persecution of Donald Trump would stop immediately," he said. "But that's not what I will do."

Despite Trump's reputation for harshly criticizing rivals, Pence and other potential GOP contenders have been increasingly brazen in their willingness to take on the man who remains a dominating force in the Republican Party, despite his actions on Jan. 6, when he stood by as a mob of his supporters ransacked the Capitol and tried to halt certification of Biden's win.

The former White House partners also campaigned for rival candidates in Arizona Friday, while Pence's former chief of staff, Marc Short, recently testified before a federal grand jury investigating the assault on the U.S. Capitol.

Short was at the building that day as Pence fled an angry mob of rioters who called for his hanging after Trump wrongly insisted Pence had the power to overturn the election results.

Pence has repeatedly defended his actions that day, even as his decision to stand up to his boss turned large swaths of Trump's loyal base against him. Polls show that Trump remains, by far, the top choice of GOP primary voters, with Pence far behind.

That contrast was on display Tuesday as Trump spoke before an audience of hundreds of cheering supporters gathered for the America First Policy Institute's two-day America First Agenda Summit. The group is widely seen as an "administration in waiting" that could quickly move to the West Wing if Trump should run again and win.

The event had the feel of a Trump White House reunion -- but one without Pence.

Pence, meanwhile, received a friendly -- but less exuberant -- welcome from the students, who struggled to break into a "USA!" chant.

In his remarks, Pence repeatedly touted the "Trump-Pence administration." But the first question he received during a brief question-and-answer session that followed his speech was about his growing split with Trump, which is particularly stark given the years he spent as the former president's most loyal sidekick.

Pence denied the two "differ on issues," but acknowledged, "we may differ on focus."

"I truly do believe that elections are about the future and that it's absolutely essential, at a time when so many Americans are hurting and so many families are struggling, that we don't give way to the temptation to look back," he said.

Pence has spent recent months delivering policy speeches, traveling to early voting states and penning a book that Simon & Schuster announced Tuesday would be titled "So Help Me God" and published in November. The publisher said the book would, in part, chronicle "President Trump's severing of their relationship on January 6, 2021, when Pence kept his oath to the Constitution."

Trump, meanwhile, has spent much of his time since leaving office spreading lies about his loss to sow doubt about Biden's victory. Indeed, even as the House Jan. 6 committee has been laying bare his attempts to remain in power and his refusal to call off a violent mob of his supporters as they tried to halt the peaceful transition of power, Trump has continued to try to pressure officials to overturn Biden's win, despite there being no legal means to decertify it.

The America First Policy Institute is one of several Trump-allied organizations that have continued to advocate for his priorities in his absence. In addition to the summit, the group has been making preparations for another possible Trump administration, hoping to avoid the early chaos of Trump's first term by "making sure we have the policies, personnel and process nailed down for every key agency when we do take the White House back," said AFPI president Brooke Rollins, who previously served as head of Trump's Domestic Policy Council.

While the organization was once dismissed as a landing zone for ex-Trump administration officials shut out of more lucrative jobs, it has grown into a behemoth, with an operating budget of around $25 million and 150 staff, including 17 former senior White House officials and nine former Cabinet members overseeing nearly two dozen policy centers.

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AP writer Lisa Mascaro contributed to this report