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LONDON — U.S. Secretary of State Rex Tillerson has been working his diplomatic muscles this week shuttling between Qatar and Saudi Arabia, trying to end a month-long rift among some of America's most important Mideast allies.

Saudi Arabia, the United Arab Emirates, Egypt and Bahrain severed relations with Qatar and cut air, sea and land routes with it more than a month ago, accusing Doha of supporting extremist groups.

The four anti-Qatar countries issued a 13-point list of tough demands, primary among them that Doha cut ties with Islamist groups — particularly the Muslim Brotherhood.

Qatar's foreign minister, Sheikh Mohammed bin Abdulrahman Al Thani, shakes hands with U.S. Secretary of State Rex Tillerson in Doha on Tuesday. Naseem Zeitoon / Reuters

But Qatar, a tiny, energy-rich nation, has rejected the demands, saying that agreeing to them wholesale would undermine its sovereignty.

The United States and Qatar signed an agreement to fight the financing of terrorism while Tillerson was in Doha on Tuesday. America's top diplomat then jetted to the Saudi city of Jeddah to sell the Saudis and its allies on the deal, and then he was back in Qatar on Thursday.

Related: Trump Credits His Middle East Trip for Saudi-Led Diplomatic Break With Qatar

So what is it about the Muslim Brotherhood, an 89-year old group that mixes politics with Islam, that is seen as a threat by several Arab states?

What Is the Muslim Brotherhood?

It is a Sunni political Islamist group founded by Islamic scholar Hassan al-Banna in Egypt in 1928.

It promotes running society along Islamic principles and has tended to pursue power through the ballot box and a gradualist approach, but some offshoots have espoused violence.

Although it is an international organization, different countries, such as Egypt and Jordan, have their own branches. Many groups draw inspiration from the Brotherhood, but coordination among the affiliates is loose.

Political groups influenced by the Muslim Brotherhood include Tunisia's Ennahda party, which has a majority in Parliament, and Turkey's Justice and Development Party, led by President Recep Tayyip Erdogan.

Strong representation of members from the professional and clerical classes within the Brotherhood has given it organizational clout and, at times, the attention of powerful backers. The Brotherhood's most famous slogan is "Islam is the solution."

Their English-language website says they are "a group established to promote development, progress and advancement based on Islamic references."

Saudi Crown Prince Mohammad bin Salman bin Abdel Aziz al-Saud and U.S. Secretary of State Rex Tillerson at a meeting in Jeddah, Saudi Arabia, on Wednesday. Saudi Press Agency

How powerful is the Brotherhood?

The Muslim Brotherhood is relatively weak given that it is banned in several Arab states, but its doctrine of political Islam has been hugely influential.

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The group's fortunes have fluctuated wildly since its formation. For example, it rose to power in Egypt after the 2011 Arab Spring revolutions, but it was deposed shortly afterward, and its leaders were rounded up and imprisoned.

It was once an important political party in Syria, but it was crushed in the 1982 Hama massacre, when former President Hafez al-Assad (father of the current president, Bashar al-Assad) killed thousands of Brotherhood members and others to quell an Islamist uprising.

The organization is also under pressure in Jordan, although its charitable activities have made it part of everyday life and won it parliamentary seats in elections last year.

"From being in a position where they had a member as the president of Egypt, they're now being pushed back heavily in Egypt and clamped down upon in other countries," said Tim Eaton, Middle East expert at the Royal Institute of International Affairs in London, also known as Chatham House.

"Generally, there was a lot of talk of 2011 ushering in Islamist governments in the Middle East — and that hasn't come to pass," Eaton said.

How does it operate?

The Brotherhood mixes political activism with social and charitable works that garner public support.

Members say they accept democracy and disavow violence, although critics point to their ideological association with the Palestinian group Hamas, which is considered a terrorist organization by several Western powers, including the United States.

The Brotherhood's detractors in the Middle East have been lobbying hard in the West for it also to be labeled a terrorist entity — but many Western allies have seen that as an overreach. And the Brotherhood insists that it is a completely separate organization from Hamas.

"Although Hamas does indeed espouse the ideology of the Muslim Brotherhood ... it does not have any shared operational or administrative functions therewith," Ibrahim Mounir, the group's most senior official in the Britain, said in evidence presented to a British parliamentary investigation of the group last year.

Related: U.S. Ambassador to Qatar Steps Down

Jordanian children chant anti-Israel slogans during a Muslim Brotherhood protest in front of the Israeli Embassy in Amman, Jordan, in July 2014 to condemn an Israeli bombing of Gaza. AP file

Why do some countries see the Muslim Brotherhood as a threat?

Different Arab states have different reasons for their hostility, but broadly speaking, well-organized political movements with popular support are always seen as threats to the entrenched autocracies represented by Saudi Arabia and its allies.

The 2011 Arab Spring protests sharpened those fears after the Muslim Brotherhood and Ennahda gained political power in Egypt and Tunisia, respectively.

The Brotherhood's transnational nature and opaque links among affiliates have also stoked suspicion among officials wary of foreign meddling.

In the case of Saudi Arabia, the group's cloudy mixture of politics and Islam is anathema — the Saudi clergy has control over religious matters on the condition that it does not interfere with the ruling monarchy's control of politics, the economy and defense.

And Saudi antipathy toward the Muslim Brotherhood is not new. In 2002, a Saudi minister described the group as the "source of all evil" in the kingdom.

Saudi Arabia, a regional heavyweight and home of Islam's two holiest sites, may also have misgivings about its tiny neighbor Qatar's forging an independent foreign policy.

"We believe the main issue is not about terrorism. The main issue is opposing differences and the way to shut the other voice," Qatar's foreign minister, Sheikh Mohammed bin Abdulrahman Al Thani, said in London this month.

"Maybe they are looking at Qatar [and thinking] that it's punching over its weight," he added.

In Egypt, where the Brotherhood has founded schools and hospitals, the government believes the group has "made use of poverty for political gains," Egypt's minister of social solidarity, Ghada Waly, was quoted in U.S.-based Foreign Policy magazine as having said.

The Brotherhood is also viewed with mistrust by secularists and religious minorities who fear persecution under its rule.

Female Egyptian supporters of the Muslim Brotherhood hold up portraits of Egypt's ousted president, Mohamed Morsi, during a rally in Cairo in July 2013. Mohamed El-Shahed / AFP - Getty Images file

Why doesn't Qatar see the Brotherhood as a threat?

Qatar is the richest country in the world per-capita, and therefore the charitable works that have won the group support elsewhere in the Middle East have no traction in the small Gulf peninsula.

"The Brotherhood's lack of penetration in Qatar is also explained by its inability to perform its usual social functions," David Roberts, an expert on the Gulf region at King's College London, wrote in an article for the Middle East Policy Council, a U.S.-based think tank.

"Running local hospitals or schools, typical Brotherhood activities elsewhere in the region, is popular but inevitably undercuts the state's legitimacy. Qatar is a wealthy ... state that provides for its citizens' every need," Roberts added.

Only around 300,000 of Qatar's population of 2.5 million people are citizens, and officials boast that a system of "consultative monarchy" gives Qataris enough say over how the country is run.

In defending its backing of the Brotherhood, Qatar argues that demonizing dissident groups risks driving them toward violent Islamic radicalism.

Qatar has hosted several senior Brotherhood figures, given television airtime to the hugely influential Brotherhood-associated cleric Yusuf al-Qaradawi, and championed the Brotherhood during the Arab Spring revolutions through the Doha-based TV news network Al Jazeera.

Experts also say Qatar's courting of the Brotherhood ties in with the small country's outsize diplomatic efforts to boost its global influence.

"In short, the country maintains a vast network of relationships with states large and small to avoid putting all its eggs in the GCC's basket," the U.S.-based geopolitical analysis firm Stratfor said in an analysis last week, referring to the Gulf Cooperation Council. "But its activities don't always suit its neighbors in the bloc."