A 22-year-old man has been arrested after an offensive Facebook page was set up following the deaths of PCs Nicola Hughes and Fiona Bone.
Several similar Facebook sites were set-up within hours of news that the two officers were shot dead after answering a routine call in Manchester.
Comments on one of the pages praised Dale Cregan, who has now been charged with murdering the two women, as a 'hero' who should be awarded an OBE and suggested more officers should be killed.
One user who posted to one of the pages refered to himself as 'bullet-tooth tony' and wrote: "dale cregan u legend".
The posts attracted widespread condemnation from police colleagues of the shot officers and from the general public.
In a statement to journalists, Assistant Chief Constable of Greater Manchester Police, Garry Shewan, said: "Tuesday's tragedy has touched many people's lives.
"As the public have been watching in from the outside, they'll have seen the hurt and the pain this tragedy has caused police and the public and those who support policing in Greater Manchester.
"Social media has given people the facility to say how they feel about the loss they've experienced but there are also a number of people who use social media in a very malicious and offensive way.
"We've become very aware that derogatory and offensive comments have been posted on social media websites.
"We received reports of at least one Facebook account had been set up with derogatory comments about the tragedy of Tuesday's events and last night, Merseyside police arrested a 22-year-old male from the Netherley area of Merseyside."
The 22-year-old was released on bail on Thursday pending further inquiries.
The overwhelming message on social media sites remained one of support and a Facebook page was set-up by members of the public.
A Twitter hashtag #coverforgmp was also started, with more than 2,700 police officers from forces around the country offering their time to allow GMP officers and staff to attend the funerals.
Thousands of people left messages of support and paid tribute to Fiona Bone and Nicola Hughes on an online book of condolence.
Greater Manchester Police Chief Constable, Sir Peter Fahy, laid flowers at the scene, alongside fellow officers.
He told reporters that he had spent the day with the families of both officers.
"They talked about how Fiona went to a particular incident involving an elderly lady who'd suffered a crime, and Fiona went back off-duty with flowers and visited the woman a number of times to make sure she was alright."
He said her partner had told him that they were "looking forward to a life together".
He added: "I've also been with Nicola's family today, they talked about someone who loved karate. Although she was very small in stature she was a real fighter.
"On shift, if there was a big fence to be climbed she tended to be the one who was picked up and thrown over the fence."
He said she once got into a house through a dog flap because she was so small.
By Thursday morning, there had been postings from 33,000 members of the public and colleagues following the gun and grenade attack in Tameside on Tuesday morning.
Some messages were from people as far away as the United States. Craig Martin, from Wisconsin, said the loss of the two officers was of "tragic proportions".
He wrote: "That these two fine Officers demonstrated, through their service and example, professionalism, dedication and commitment is obvious to all".
Jon, an employee with the London Ambulance Service, wrote: "RIP dear colleagues. Our prayers are with you and condolences to your loved ones".
Greater Manchester Police officers have also expressed sadness at the loss of their colleagues. Vicki, a GMP officer, wrote: "Goodbye sisters. We're very proud of you and will miss you".
In response to the messages, Sir Peter posted a video message on YouTube, thanking those who contributed to the book which the force had set up.
"Since the awful deaths of our two police officers, we've received thousands of messages of support and condolences from the public and a huge number of hits on the Facebook site," he said.
"We'd like to say a huge thank you to the public for this great show of support. I can't tell you how much it means to us at this difficult time."
The brutal nature of the killings has shocked Manchester, a city still trying to shake off its gangs-and-gun-culture legacy of the 1990s.
Manchester United players wore black armbands during their Champions League match against Galatasaray at Old Trafford on Wednesday in a tribute to the officers.
Manager Sir Alex Ferguson said: "We are all deeply saddened by the tragic loss of two young policewomen protecting our community."
The Queen also paid tribute to the officers. A statement issued by the Lord-Lieutenant of Greater Manchester's office, said: "Her Majesty expressed her sincere condolences to the families of PC Nicola Hughes and PC Fiona Bone at the tragic loss of their lives."
Home Secretary Theresa May cut short her holiday to travel back to the UK to visit GMP's force headquarters. She said Fiona Bone and Nicola Hughes were "incredibly brave" officers.
"The murders were savage acts of pure brutality," she said. "They've left grieving families and they've devastated the Greater Manchester Police force.
"These were brave and talented offices, going about their duty, protecting the public and fighting crime and they were incredibly brave".
Miss Bone lived in Sale with her partner Clare and daughter Jessie. She had served Greater Manchester Police for five years.
In a statement, her family remembered their "wonderful daughter, sister, partner and parent".
"She leaves behind her Mum, June, Dad, Paul and her sister Vicky and Vicky's Fiancee Mark. Her partner Clare and daughter Jessie," it said.
"Our thoughts are with Nicola's family along with Fiona and Nicola's fellow officers at Tameside police station."
Miss Hughes' family also issued a statement, which read: "Nicola was our only daughter and a beautiful child. She was always happy with life and lived for her family.
"She had an infectious personality and sense of humour and was a very caring and loving girl. When she left the house this morning she was going to the job she loved".
The two PCs were killed as they responded to a routine call at Abbey Gardens in Hattersley shortly before 11am on Tuesday.
The post-mortem examination for both officers has concluded they died as a result of gunshot wounds.