Festering hostilities between rival Christian factions in northern Lebanon have erupted into a wave of kidnapings and alleged executions, dealing the government a setback and posing a challenge to Syrian peacekeeping forces.

A previously unknown group calling itself the "revolutionaries of the North" and believed to be an offshoot of the right-wing Christian Phalangist Party, issued a communique today in which it claimed responsibility for "executing" five followers of former Lebanese president Suleiman Franjieh. Franjieh hails from the northern town of Zghorta.

TheVoice of Lebanon radio, organ of the Phalangist Party, later reported that Franjieh supporters had surrounded the village of Samar Jbeil, also in the north, and killed three people.

The communique said the executions, carried out at daybreak, were to "mete out justice to the Franjieh clan and those who have cooperated with the Syrians or participated in military operations against innocent civilians."

The inter-Christian feuding, in the fierce tradition of the mountain-dwellers of northern Lebanon, has shown no sign so far of escalating into a wider sectarian conflict involving any of the Lebanese Moslem or Palestinian combatants in the 1975-76 Lebanese civil war.The Palestinians, especially, are too preoccupied with their new "diplomatic" gains to get involved, observers said.

The feuding represents an affront to the Syrians, who have close ties to the Franjieh clan. But growing domestic problems in Syria between Sunni Moslems and the ruling Alawite Moslem minority make it increasingly difficult for Syrian forces to intervene.

Syrian peacekeeping troops in Lebanon battled the Phalangists and other right-wing Christian militiament sporadically through 1978.

Bickering between the Franjieh and Phalangist supporters first surfaced when the former Lebanese presient dropped out of the Lebanese Front, a coaliton of Christian factions including the National Liberal Party led by former president Camille Chamoun and the Phalangist Party led by Pierre Gemayel.

Franjieh, who at the time said he did not approve of the Front's policies, allied himself with the Syrians, who have a force of 22,000 troops in Lebanon as part of an Arab League-sponsored force deployed at the end of the civil war.

The underlying conflict between Franjieh supporters and Phalangists and their sympathizers in the north arose from an ideological split over allegiance with Syria and competition for influence in northern Lebanon.

Phalangist Party officials claim that in the spring of 1978, following the "defection" of Franjieh from the Lebanese Front, seven party members were killed at the hands of Franjieh supporters. The most prominent was Youssef Bayeh, who was one of the "highest Phalangist officials in charge of north Lebanon," according to a rightist official.

A week after Bayeh was killed in June 1978, gunmen believed to be Phalangist militiamen stormed the house of parliamentary deputy Toni Franjieh, son of Suleiman Franjieh. In a predawn attack, the unit killed Toni, his baby daughter, Egyptian-born wife and about 30 members of his entourage in the summer resort town of Ehden.

A mutual vendetta since then, which flared up with the detention by Phalangist gunmen Monday of as many as 50 Franjieh loyalsits, continually has thwarted government efforts to restore security in Lebanon.

The latest killings caused Lebanese President Elias Sarkis to call an emergency Cabinet session todday attended by top security officials.

"Every time a glimmer of hope appears on the horizon," Prime Minister Selim Hoss said afterward, "something happens to disrupt the will of the state.