"I remember," said Bob Killian, standing in the Summit Tavern, surrounded by old family friends and supporters from the Irish section of Hartford called Frog Hollow. "I remember when we used to sing that song. The Hollows come out every night. They rumble and tumble and fight."

"That's right, Bobby," shouted Bud Mann, the white-haired proprietor, who has been Hartford's city treasurer for almost half his 66 years of life. "But I've never seen a fight like this one in my whole life."

On the wall behind Mann was a blown-up newspaper photograph of Irish Bob Killian extending his hand in greeting to Ella Grasso, elected four years ago as the first woman and the first Italian-American governor in Connecticut's history. She is gazing at him addly, her hands clasped before her. According to a story believed by everyone in the Summit, she was cursing him in a stage whisper the photographer could overhear.

That's the way it is this year in Connecticut, as Gov. Ella Grasso faces Lt. Gov. Robert K. Killian in a Sep. 12 Democratic gubernatorial primary the likes of which this state has rarely seen.

The betting is that Grasso will beat her reluctant 1974 running mate in the primary, although some senior Democrats think the animosity toward her is great enough to fuel an upset. The real fear among the state's Democratic leaders is that the wounds of this battle may make it hard for either Grasso or Killian to withstand the November challenge of young Rep. Ronald Sarasin, who heads a GOP "dream ticket" assembled by state Republican chairman Frederick K. Biebel.

All this is part of what many here see as the unraveling of a system of Democratic dominance maintained by John M. Bailey, who was state Democratic chairman for 29 years before his death in 1975.

It was Bailey, in 1955, who pushed through the legisalture the "challenged primary law," which required a candidate to receive 20 percent of the delegate votes at the state convention in order to seek a primary against the convention-endorsed candidate.

Republicans held one gubernatorial primary under the new law, but Bailey was always able to prevent such bloodletting within his party. Coincidentally or not, the Democrats won the governorship in every election except 1970.

That may explain why Grasso leaned over to a reporter at the tumultuous Democratic convention last month and said, in her best Mt. Holyoke manner, "John Bailey, thoushouldst be living now. We have need of thee at this hour."

That convention was replete with fronies and intimations of political disaster. Four years ago, when Bailey privately favored Killian for governor and Grasso for lieutenant governor, the lady - who had patiently served a 22-year apprenticeship in the lower officer - firmly informed her longtime friend and mentor that it was her turn for the top of the ticket.

She formed an alliance with Nick Carbone, the street-smart and ambitious president of the Hartford City Council, who had chafed under Killian's rule as Hartford Democratic chairman. With Carbone's help, she beat Killian 2 to 1 in the battle for the key Hartford delegation. At that point, Bailey intervened and persuaded a reluctant Killian to accept the lieutenant governor's nomination. Tr for add four

According to both rivals, the political marriage Bailey arranged was not made in heaven. "She shut the door on me," Killian says. "At first, she used to call me for advice, early in the morning and late at night. But if I said anything but "You're brilliant, Ella' said resented it. I'm too old for that."

"He never raised a criticism or a protest in three years," Grasso replies. "In fact, he never did a thing." Echoes her husband. Tom, a retired school principal who, since Bailey's death, has been her principal political adviser: "Killian had a great opportunity to contribute to this administration, but he didn't want to seize it."

Killian's acknowledged ambition for the governorship was whetted last year, when reports circulated that Grasso was hoping for an ambassadorial appointment from Jimmy Carter. But none was forthcoming, according to Connecticut Democrats, because Carter had not forgiven Grasso's diehard support for his nomination rival, Sen. Henry M. Jackson (D-Wash.).

Killian's opportunity for revenge came about this year because Grasso, who has seen almost two dozen top appointees quit and many original political backers defect,allowed an open breach to develop with Carbone - the man who had broken Killian's ambitions in 1974. Carbone blames the split with the governor on her failure to keep campaign commitments on urban aid. She says it resulted form her refusal to let Carbone name key commissioners in her administration.

In any case, it was Carbone who gave Killian just enough convention votes to force a primary. In the tumult of the convention hall, Grasso supporters tried desperately to pressure delegates away from Killian.

The outraged lieutenant governor stopped that tactic only by threatening to run as an independent in the general election if he were denied a primary. But the bitterness toward what he called "strong arm tactics" increased when Grasso accused Killian of playing "gutter" politics and later refused to shake hands with him.

Killian's reputation was that of a very ordinary machine politician and no liberal. But he has moved to the left in this campaign, seeking labor, urban and minority support. He accuses Grasso of compromises with Republican big business leaders, some of whom have been serving as advisers to her administration.

A series of pointed but funny Killian radio ads charge Grasso with using state contracts to "pay back" political favors and of temporizing on tough proglems. Her road construction program, an ad says, is wonderful "if you love cliffs."

Republicans, understandably, love the Killian ads.

Grasso , who makes little effort to conceal her contempt for Killian or her irritation at his challenge, talks about her record of attracting 260 new businesses to the state and bringing it back from a $70 million deficit to a $95 million surplus.

Her supporters pounced on a Killian suggestion in their first debate that he might support a new tax on stocks, bonds and jewelry - a bit of populism that longtime Killian backers on Carbone and concede may cost Killian votes.

Behind the ostensible issues, the basic question is how much the political mistakes Grasso has made since Bailey's death may have fed the ethnic enmitics that are never far below the surface of Connecticut politics.

The Grasso-Killian fight is, in large part, a struggle for dominance between the Italian-Americans, who are enjoying their first governor, and the Irish-Americans, who preceded them to power.

It is compounded by the fact that Grasso, breaking with Bailey's tradition of handpicking the entire state ticket, left the choice for the lesser offices to the convention. She wound up with egg on her face - a ticket including neither a Jew nor a Pole among its six members.

Exploiting this weakness, Republican Chairman Biebel pulled off a coup at the GOP convention worthy of Bailey at his best. With Sarasin, a French-Portuguese Catholic who has served three terms in the House from a normally Democratic district, showing as Grasso's strongest potential challenger, Biebel persuaded Sarasin's main rival, Lewis B. Rome, a highly regarded Jewish state senator, to accept the lieutenant governor nomination. Then he balanced the ticket with an Italian-American, a Yankee, and two women.

Some GOP polls show Sarasin narrowly ahead of Grasso already, with Killian presumably the weaker general-election candidate. "I always knew there would be a period of chaos after John Bailey died," one leading Democrat said. "But this is worse than anything I ever imagined."