Gen Abdelaziz has said he wants to tackle the roots of extremism
Three men have each been sentenced to 10 years in prison in Mauritania for reportedly belonging to a group linked to al-Qaeda's North African wing.
The sentences are said to be the harshest since the country introduced anti-terrorism laws three years ago.
The men were arrested earlier in the year in Mali and then extradited to face trial in Mauritania.
Al-Qaeda in the Islamic Maghreb is accused of staging attacks across North Africa in recent years.
A fourth man, who is already serving a three-year sentence for similar offences, was given five years for recruiting and financing terrorist activities.
Sources at the court said the men are able to appeal against their sentences.
An internet message purported to be from the North African wing of al-Qaeda in August urged Mauritanians to take up arms against General Mohamed Ould Abdelaziz and the other military officers who toppled the country's first democratically-elected president.
Since taking power, Gen Abdelaziz has said defeating extremism was one of his main priorities.
Al-Qaeda in the Islamic Maghreb is largely based in Algeria but has been blamed for attacks in Mauritania.
The country's former government said it killed four French tourists last December, an incident that prompted the cancellation of the Paris-Dakar car rally.
It also blamed the group for attacking the Israeli embassy in the capital, Nouakchott, in February, as well as attacks in Algeria and Morocco.