What is the order of any element is a group of order three?
Can any non-identity in the group generate the group?
Are cyclic groups Abelian?
I have some modern algebra homework due tomorrow (and a midterm in there, ) and I've done all of the homework except this problem. I can't for the life of me figure out why a group of order 3 is abelian. I don't see the correlation between a groups order and whether or not a group is communitive.
Do any of you guys have any idea how to show this? Or could you just give me some hints that will get me started?
Thanks.
I kind of figured it was, but I wasn't positive, as it hasn't really been introduced to us yet. I found some proof explaining why it is, however and convinced myself of it.
So, it's because every element in a group of order 3 has order 3, and the entire group can be generated by a non-identity element? Because it can be generated by a non-identity element, the group is cyclic, and every cyclic group is abelian?
Sorry if this seems like a really dumb question. I'm just really burning out on math. I have 2 math midterms tomorrow, and one next week, and I've been cramming hard core for a while now for it.
Yes, I would say you learn that theorem 2 months into that course. It is called Lagrange's theorem (actually a consequence of it). If you never learned that theorem we can construct, as topsquark, suggests a table for group of order 3 and show that it must be unique.