Do you have this ... cryptographers number? Asking for a friend Groups are one of the more basic yet powerful math constructions/areas of study. Just 3 simple axioms and quite a bit follows. One cool thing that can help conceptualize them a bit, is it can be shown All finite groups can (say of order N) can be embedded in the set of permutations of N numbers (the set of functions that mix 1-1 a set of N objects, this has N! functions in it, and the group operation is composition of functions. So you can look at groups as set of permutations (that's where that "invertible" axiom is needed, permutations being 1-1 are invertible functions) that are closed under this operation. Neither here or there, but I think knowing this helps understand the motivation or interest behind them). Abelian groups are those that are commutative, order doesn't matter, and all abelian groups are classified nicely as "direct products" of cyclic groups (eg Z3 x Z5 is an abelian group of order 15). Cyclic groups are just an even narrower set of groups that have just one multiple in these direct product things. In elliptic curve setting, we construct a convoluted operation (the weird adding of points on a curve) and then show its a group. If it turns out this group has say prime order, it must be cyclic (easy enough to prove this), so we get that nice presentation using generators. This is rough and a couple things I could smooth over but don't wanna