Ok, here is the situation. I am working on a take home in which I have this problem involving the order of a group. Let: G be the group of all non singular matrices (matrices that you can invert) and the operation on the group is regular old matrix multiplication. a = a matrix: |0 -1| |1 0| b = another matrix: |0 1| |-1 -1| now, a has order four. This means that if you multiply a by itself a couple of times, you eventualy get the same three elements over and over again. b has order three. However, ab has infinite order. This is a concept that I have a little bit of trouble getting my head around. If we are talking about cyclic groups, ( I assume that we are, but If we arent tell me how) then a group with infinite order seems rather rediculous because there is no way that you can keep multiplying till you get tired, your mother calls you downstairs for dinner, or you reach infinity (in which case you have lost it) All in all I am a little confused with this problem and I wanted to know if anyone could point me in the right direction. Dont do the whole thing, because I need the mental practice. I thank you in advance for your help.