I think the suggestions made by kcarrn are very sound. Unfortunately, it is difficult at this age to know what may be the underlying cause of these behaviors. It can come from many different sources, both internal/genetic and external/environmental. The most likely things to first rule out are: 1) ADHD and 2) a mood disorder (depression with associated mood irritability). An important piece of information in this regard is whether anyone else in the family suffers from these problems, especially in first and second degree relatives (i.e., mother, father, siblings and grandparents, aunts & uncles). Professional help, including comprehensive psychological testing, and ongoing supportive counseling can also be very helpful in sorting things out. Tough stuff to deal with but there is help and it is so important that you are open to looking at these problems at such an early age and you're not attributing it to simple willful misbehavior. There are some good children's books on anger and anger management, so I encourage you to talk to your kids about working on these behaviors and, as mentioned about, trying to identify triggers to the outbursts. Good luck.
You are also describing my six year old son. I mean a mirror image. I am at my wits end also. He has run out of the school with the gym teacher chasing him. He also has a 4 year old brother that he loves but if he gets at him, he will become violent with him. At the same time, if you make his little brother upset, then he will turn that anger/aggression on that source. He gets really upset when he has misplaced something or gets something incorrect. We are trying to find out what "triggers" the Jekyll and Hyde transformation in him. Naturally, if he is tired, he is more prone to anger. He really loves animals though. He can recite the oddest facts about whales, lions, etc. But only about animals. He hates to color because he gets mad when he goes outside the lines.
We had to have a meeting with school personnel because he was getting angry in class so much and having to go cool off, that he is missing about 2 hours of class each day. Not to mention the amount of time that takes from the teachers ability to attend to the other children.
The school counselor suggested we try to set a really strict daily routine for him, so that he knows what to expect and what is expected.
He seems to respond slightly to these "set" routines. The same pre-bedtime routine, the same "wake up" routine. He gets to school at the same time each morning. It's to the point that the school principal comes and has a pep talk with him each morning. She missed one morning and he had a bad morning until he got sent to her office. Then he was ok the remainder of the day.
We have taken him to private counseling in the past and I think we are going to start back. At the time he was 5 and we were told that he was too young to get any definitive answers. And that most likely it was just temper tantrums. But I don't call threatening to kill your whole class, picking up a chair at your teacher just a temper tantrum.
You could maybe try counseling or the strict routines to see if it helps. A