Police say they are investigating a "network" over Monday night's attack at Manchester Arena, in which 22 people were killed.
Home Secretary Amber Rudd said: "[Monday's attack] was more sophisticated than some of the attacks we've seen before, and it seems likely - possible - that he wasn't doing this on his own."
Hamid El-Sayed, who worked for the UN on tackling radicalisation and who now works at the University of Manchester, said Abedi had a "really bad relationship" with his family and his parents had tried but failed to keep him on the "right path".
Former classmate of Abedi's told the BBC that he was a "very jokey lad" but also "very short tempered" and would get angry at "the littlest thing".
The man, who did not want to be identified, said Abedi was "away at random times throughout the year" but he did not know if he was abroad or playing truant because he hung around "the wrong crowd and was very, very gullible".
He said that, before leaving the school in 2011, Abedi became "more and more religious" and that this might explain why he cut ties with former classmates.
Soldiers join police officers patrolling in London Prime Minister Theresa May said soldiers are being placed at Buckingham Palace, Downing Street, embassies and the Palace of Westminster to support armed police in protecting the public.
Military personnel may also be seen at other events over the coming weeks, such as concerts, Mrs May said, in what she is calling a "proportionate and sensible response".
The highest threat level has only been reached twice before: in 2006 during an operation to stop a plot to blow up transatlantic airliners and a year later when security chiefs were hunting for the men who tried to bomb a London nightclub and attacked Glasgow Airport.