My daughter (who is now 22 months old) went through exactly the same thing, and she too is an only child. Thus, the problem would mainly occur only at school, but occasionally at home. She would either bite others or bite herself (enough to leave marks!!!)
Some ideas behind what is causing the problem:
1. Frustration at not being able to communicate well
Suggestion: work on teaching non-verbal communication (sign-language), this helps them express what they want, when they don't have the verbal ability or vocabulary to do so.
2. Showing anger or simply curiosity towards other children (the toy jealousy thing)...
Suggestion: At school, have the teachers sternly say "No-no" and put her in an isolated area(by this I mean away from the other kids) as soon as the incident happens. If all of the other kids are at the table, have them sit her on the play area away from others. Certainly a place where the care-givers can still watch her (so not in complete isolation). This will remove the immediate threat to the other children and hopefully express to your child that what she is doing is not acceptable.
3. Teething
There is also a possibility that she is teething. Therefore your child will bite to relieve the discomfort of teething (and sometimes, the other kids are the most accessable object). So we kept a chilled teething ring at daycare. If our child bit another child, tried to bite another child, or looked as though she was about to bite another child... we would have the watchful daycare personel give her the teething ring. If it was a teething problem, it would help soothe it. However, it is an anger or frustration problem, it teachs your child that there is only a certain object (the teething ring) that is okay to bite.
Hope this helps or gives suggestions!
C.