Britain's great bobby debate: what is their role?

The scene is being set for a sweeping review of the role of the police in British society as crime figures soar and law and order becomes an intensely political issue.

Key ingredients in what is already being seen as a great debate include:

* The surprise resignation of Britain's top policeman amid suggestions that he was not getting strong enough backing from the government.

* Insistent calls from policemen on the beat and Conservative politicians for a return to the death penalty as punishment for murder.

* New crime statistics showing that mugging and burglary are rising sharply.

* Pressures on the home secretary, William Whitelaw, from other members of Prime Minister Margaret Thatcher's Cabinet to give the police greater powers for dealing with crimes of violence.

* A new ethnic factor in assessments of crime, with London's police for the first time issuing statistics showing that in some categories of crime, black people are more frequently responsible than white people.

In some ways the most striking of these developments was the decision of the metropolitan police commissioner of London, Sir David McNee, to step down after only five years in the office.

Sir David, a tough Glaswegian whose nickname is ''the hammer,'' denied that he and Mr. Whitelaw had clashed over police policy for London, but close associates suggest otherwise.

The London police came in for heavy criticism of their handling of rioting in the inner suburb of Brixton last year. The criticism caused resentment among police on the beat, who claim they are hampered in their handling of crime where minorities are crowded into poor housing areas.

Only days after Sir David resigned, he declared ''get off our backs and allow police officers to carry out their job. The job is to stop the upward trend in crime and halt growing lawlessness in society.''

Into Sir David's shoes as London's police chief steps Sir Kenneth Newman, a former head of the Royal Ulster Constabulary. A tough disciplinarian, he has said that modern police forces need to be ''hard but sensitive.''

The mood of policemen on the beat is militant over capital punishment, abolished in the 1960s. Their professional body, the Police Federation, has taken display advertisements in leading newspapers calling for the restoration of hanging.

There have been a number of deaths of policemen on duty in recent months. And these have helped to stoke the anger of the British bobby. Police spokesman Jim Jardine has called frequently for a ''war on the criminal'' and for policemen to be given more powers to deal with lawbreakers.

The debate about law and order entered the political arena earlier this month when the London police for the first time issued crime statistics giving a breakdown of criminal activity on ethnic lines. It tended to show that black people were more likely to be the attackers in street crime.

There was an immediate uproar in the House of Commons, where the Labour opposition complained that drawing ethnic distinctions in crime figures was pernicious. Police in London defended the figures on the grounds that they gave a true picture of what was happening.

But the Labour opposition leader, Michael Foot, broadened the debate by asserting that the root of crime lay in the state of the British economy. With unemployment running just below 3 million, and a higher proportion of blacks out of work than whites, it was inevitable for the figures to seem alarming, he said.

Prime Minister Thatcher denies there is a direct link between unemployment and crime. Mr. Whitelaw, who is opposed to capital punishment and enjoys a liberal reputation for his attitude to law enforcement, has been heavily criticized by some of the prime minister's associates for failing to do enough to cope with the crime wave.