Hello, I just started doing groups in my algebra class and I am struggling with the abstraction of it as usual. Here is how my class defines a group: A group is a nonempty set G with a binary operation "o" such that for all of x,y,z which are elements of the group the following holds: (P0) If x and y are elements of G, then x o y are elements of G. (P1) x o (y o z) = (x o y) o z (P2) There's a u which is an element of G such that u o x = x = x o u (P3) For every x which is an element of G there is an x-1 such that x o x-1 = u and x-1 o x = u. Here is my question: Which of the following are groups? Justify your answer. iii) The set of all rational numbers except -1, with the operation o defined by x o y = x + y + xy. iv) The set of all integers that are multiples of a fixed integer d, with the operation of addition. I think I have iv). I believe it is a group. By (P0), let t = xd and u =yd where x and y are elements of Z. Then u + t = xd + yd = d(x + y). By (P1) let p = wd, t = xd and u =yd. Then p + (t + u) = (p + t) + u By (P2) let u = 0*d and t = xd then t + u = 0*d + xd = 0 + xd = xd and vice versa. By (P3) Let t = xd and w = -xd then t + w = u = w + t. Now for iii), I am not sure if I understand the situation. I think that addition by rationals results in rationals. I also think that addition of rationals are associative. But I am not sure how to apply P2 and P3 especially since I am not sure how to interpret and how to apply the axioms P2 and P3 to it. It may seem obvious to others, but it is not obvious to me (sometimes I have to read the most elementry passages many times and wait for a period of time before I absorb it). Perhaps someone who understands the situation can reword it for me? It may help me understand it better. Any help is appreciated. Thankyou.