But they are in the minority, because most bullies are themselves victimised by other children, according to the poll of 2,377 primary pupils from Hertfordshire and north London.
Boys were more likely to admit to being bullies than girls, the study found. They were also more likely to resort to verbal abuse as well as being responsible for more physical attacks.
Parents of the handful of bullies who escaped victimisation said their children had significantly fewer tummy aches, colds and coughs and were less likely to invent illnesses, Professor Dieter Wolke, of University of Hertfordshire's department of psychology found.
"It appears that pure bullies are healthy individuals, who enjoy school and use bullying to obtain dominance," says Professor Wolke, who labels these children "cool operators".
Almost half of all pupils said they suffered verbal bullying. A quarter claimed to suffer physical attacks several times a week - three times the level in German schools, according to findings presented this week at the British Psychological Society's London conference.
Fewer than 3 per cent admitted to being habitual physical bullies who beat up other children on a weekly basis.
A further 12 per cent to 16 per cent said they tormented others "frequently", although almost all were also bullied.
The researchers found that bullying in the UK, though not in Germany, was more likely to occur in small rural schools and classes.
Meanwhile, a separate study has found that Asian schoolchildren in Britain are just as likely to be bullied by children from different Asian groups as by white children.
Nearly 250 Hindu, Indian Muslim and Pakistani children in the north-west of England were asked their experiences of being bullied.
Bullying was found to be widespread with 57 per cent of boys and 43 per cent of girls reporting being victimised, Dr Mike Eslea of the University of Central Lancashire found.
Children from all three ethnic groups reported similar levels of bullying but were rarely bullied by others of the same ethnic group. However, children from different Asian ethnic groups and white children were equally likely to be sources of bullying, the study found.
Details of the Hertfordshire bullying research available from Professor Dieter Wolke, e-mail: email@example.com
Details of Lancashire research from Dr Mike Eslea, e-mail: firstname.lastname@example.org