PARIS >> An Algerian splinter group from al-Qaida has beheaded a French hostage over France’s airstrikes on the Islamic State group, in a sign of the possible widening of the crisis in Iraq and Syria to the rest of the region.

The killing of Herve Gourdel, a mountaineer who was kidnapped while hiking in Algeria, was a "cowardly assassination," a visibly upset French President Francois Hollande said Wednesday, but he vowed to continue the military operation.

"Herve Gourdel is dead because he is the representative of a people — ours — that defends human dignity against barbarity," Hollande said on the sidelines of the U.N. General Assembly meeting in New York. "France will never cede to terrorism because it is our duty, and, more than that, because it is our honor."

On Friday, France joined the U.S. in conducting airstrikes on the Islamic State group in Iraq. Two days later, the Islamic State group called on Muslims to attack foreign targets, and the response in Algeria raised the specter of attacks on Westerners elsewhere.

Gourdel, a 55-year-old mountaineering guide from Nice, was seized Sunday night while hiking in the Djura Djura mountains of northern Algeria. His Algerian companions were released.

A group calling itself Jund al-Khilafah, or "Soldiers of the Caliphate," split from al-Qaida and pledged allegiance to the Islamic State group two weeks ago. It seized Gourdel in response to the call to kill the "spiteful and filthy French." It gave France 24 hours to end its air campaign.

A video posted online showed masked gunmen standing over a kneeling Gourdel. They pledged their allegiance to the leader of the Islamic State group, Abu Bakr al-Baghdadi, and said they were fighting his enemies. The video showed the captive pushed to the ground and blindfolded before he was beheaded.

The videos from the group were similar to those from the Islamic State group, which killed two American journalists and a British aid worker in recent weeks.

"It is not the first time France has been affected by terrorist acts," Hollande told an unusual session of the U.N. Security Council chaired by President Barack Obama. "And we have never given in. Every time, we come out of these things more robust, with greater solidarity."

Obama, speaking at the same meeting, said people around the world had been "horrified by another brutal murder."

"These terrorists believe our countries will be unable to stop them. The safety of our citizens demands that we do," Obama said at the meeting, which was aimed at combating the threat posed by foreign fighters joining extremist groups.

The Algerian government called the killing of Gourdel "an odious and abject act committed by a group of criminals."

Gourdel, an avid photographer, had expressed excitement on his Facebook page about his planned camping trip in the remote mountainous region. The area, which is riddled with steep valleys and deep caves, is also one of the last strongholds of the Islamist extremists in northern Algeria that have been fighting the government since the 1990s.

The Algerian government statement said that since the kidnapping, authorities had been working to try to free him. It said it was determined "to pursue its fight against terrorism in all its forms, while guaranteeing the protection and security of all foreign nationals on its territory."

The Islamic State group claims leadership of all Muslims and has been hoping to incite additional attacks against foreigners around the world.

"That was the Islamic State’s intention, for there to be more events like this," said analyst Geoff Porter of North Africa Risk Consulting. "If there were to be any similar copycat instances, I don’t think they would transpire in Algeria, they are more likely to occur either in Tunisia or Morocco — it’s certainly a more target-rich environment."

Thousands of Tunisians and Moroccans have joined the Islamic State to fight in Syria and Iraq, and there are fears they will carry out attacks in their home countries upon their return.

The killing of a hostage actually represents a departure for radical Islamic groups in Algeria, which in the past decade have made millions from ransoms. France is also known for paying ransoms, although some hostages have been killed by their captors.

Islamic extremists have long singled out France as a special target for multiple reasons: the French military campaign against al-Qaida-linked militants in Mali, the French involvement in the NATO force in Afghanistan, and French laws banning the Muslim face veil and headscarves in public.

Hours after French warplanes struck targets Friday in Iraq, French Foreign Minister Laurent Fabius told the U.N. Security Council: "We are facing throat-cutters. They rape, crucify and decapitate. They use cruelty as a means of propaganda. Their aim is to erase borders and to eradicate the rule of law and civil society."

Nearly 1,000 French radicals have joined or are trying to join the Islamic State group in Syria and in Iraq — more than the number of fighters from any other Western country. French authorities are particularly concerned they will return home and stage attacks. Security has been boosted around the country.

The Algerian military has never been able to eliminate the vestiges of the once-powerful al-Qaida in the Islamic Maghreb hiding out in the same terrain where Algerians fought French colonizers in the 1950s. The extremists usually left civilians alone and clashed only occasionally with army patrols.

Gourdel’s killing may push the military to take care of these groups once and for all. It has sent thousands of troops and helicopters into the mountains.

Hollande praised Gourdel as a man devoted to mountain climbing who "thought he would be able to pursue his passion."

According to a presidential aide, Hollande has spoken with Gourdel’s family, and his hometown in southern France planned a vigil Thursday at the mountaineering office where he worked.

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Schemm reported from Rabat, Morocco. Angela Charlton in Paris, Karim Kebir in Algiers, Algeria, and Gregory Katz, Josh Lederman and Cara Anna at the United Nations contributed to this report.