Image copyright European Photopress Agency Image caption It was the latest in a string of random attacks on the island

A day after a man grabbed a four-year-old girl off her bicycle and decapitated her with a cleaver in broad daylight, Taiwan is in shock.

"A mother couldn't protect her child even though she was standing right near her. How can we feel assured that our children will be safe?" asked Lindy Wang, a mother of three.

On Monday night, anger poured out on the streets when crowds surrounded the suspect, a man in his 40s, as he was being led into a police station. Some punched him.

Schools were on alert on Tuesday and police officers were dispatched to a school the assailant was believed to have been headed to when he came across the girl.

The killing was the latest in a string of random attacks on the island and the third against children in the past four years.

Further heightening fears, on Tuesday, a police officer was stabbed in the head by a 28-year-old man after he approached the man to ask him why he was at a train platform with a steak knife. The officer's injuries were non life-threatening.

Image copyright AFP Image caption Angry crowds surrounded the man suspected of carrying out the attack, some trying to punch him

Taiwanese people say such crimes did not use to happen on this small island, rated as the second safest place in the world because of its low crime rate according to global surveys.

Huang Tsung-jen, the deputy director general of the National Police Agency, told the BBC: "This is not a crime problem, but a societal problem."

The suspect had a history of drug offences, was unemployed for some time and had sought treatment at a psychiatric hospital.

"The whole society's safety net has loopholes - we should be asking whether we have enough resources to help the mentally ill and the unemployed," said Mr Huang.

He has urged the public to be calm, saying Taiwan's crime prevention system is good, and its crime rate has been declining significantly. It dropped to its lowest level on record last year.

But that has done little to reassure people.

Image copyright European Photopress Agency Image caption The four-year-old girl was called 'Little Lightbulb' by her family

After the killing, legislators immediately proposed a bill requiring random killers of children to be sentenced to death.

Taiwan has come under criticism from Amnesty International and the European Parliament for maintaining the death penalty, but most Taiwanese support it, and this latest attack has made some want it used more liberally.

"In the past, our death penalty laws were very tough," said Ms Wang.

"There wasn't a long process of court appeals and executions were broadcast. But now offenders know there will be groups protesting on their behalf and they can stay in prison and be fed and housed on taxpayers' money for years."

The attack and other random violence in recent years - the murder of a girl at an elementary school, the killing of a boy in a public bathroom, a stabbing spree on a subway train that killed four, all crimes committed by young men - have also raised questions about how Taiwan's youth are being brought up.

Some believe their moral upbringing is being neglected by parents and they are put under too much pressure to succeed. Many children no longer live near grandparents or extended family, so there are fewer people playing a role in their growth.

As TV stations broadcast family videos of the latest victim, calling her by her nickname "Little Lightbulb" because of her outgoing personality, many Taiwanese fear such crimes will happen again.

Despite the outpouring of concern and vows by politicians to toughen laws, they know Taiwanese society has changed and they can no longer take for granted their safety and that of society's most innocent.