She would then be replaced by her nemesis, Khaleda Zia, of the opposition Bangladesh Nationalist Party.
Jamaat is an ally of the BNP, which is led by former Prime Minister Khaleda Zia.
The trial against Khaleda Zia, prime minister till October 2006, is yet to begin.
The protesters shouted slogans like 'Down with corruption' and 'We want Khaleda Zia to go' from their party offices.
Jamaat-e-Islami was a key partner in the former government of Khaleda Zia, a longtime political rival of Prime Minister Sheikh Hasina.
General elections are due in Bangladesh in July and Sheikh Hasina is being challenged by opposition leader and former prime minister, Khaleda Zia.
But more fuel was added to the fire by former Prime Minister Khaleda Zia, leader of the main opposition, the Bangladesh Nationalist Party.
Prime Minister, Khaleda Zia, and opposition leader Sheikh Hasina have both condemned the murder, as has the international journalists' pressure group Reporters Sans Frontieres.
This week its leader, Khaleda Zia, appointed her son, Tarique Rahman, for many the symbol of all that was wrong with the BNP's last period of kleptocratic, vindictive rule, as the party's deputy chairman.
The political scene in Bangladesh has been dominated for decades by bitter rivals Khaleda Zia and current Prime Minister Sheikh Hasina, whose electoral campaigns have sparked violent clashes and on occasions prompted military intervention.
Mr Azam lived as an exile in Pakistan and the UK but returned to Bangladesh in 1978 when the country was led by Gen Ziaur Rahman - the assassinated husband of the country's current opposition leader Khaleda Zia.
Police and witnesses said supporters of the Bangladesh Nationalist Party (BNP), led by former prime minister Begum Khaleda Zia, and its allies set ablaze about 30 buses, trucks and cars in the capital Dhaka and other parts of the country.
The government fears that, if it allows the export of natural gas, the opposition will accuse it of selling out to India, though the leader of the opposition, Begum Khaleda Zia, has actually softened her line on gas exports recently.
When Sheikh Hasina, leader of the Awami League and current prime minister, and Khaleda Zia, leader of the Bangladesh Nationalist Party (BNP), alternated in power in the 1990s, things were pretty bad, but in the past decade they have got worse.
But the trials are also seen as part of a long and bitter rivalry between Prime Minister Sheikh Hasina and the main opposition leader, former Prime Minister Khaleda Zia, who is allied with the Islamic party Jamaat-e-Islami, many of whose leaders face charges before the tribunal.
Beyond the archipelago, her ability to overcome force without resorting to violence made her a role model for an ever-lengthening line of women leaders--Violeta Chamorro, Benazir Bhutto, Chandrika Kumaratunga, Khaleda Zia, Megawati Sukarnoputri, Aung San Suu Kyi--who, like her, were thrust into public life by the violent fates that befell husbands and fathers.
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