 Let G be a group and let little G be an element of that group. You'll recall that previously we've defined the order of a group, that if you take the cardinality of the set associated to the group, we call this the order of the group. Now every element of the group itself, it creates a subgroup of itself. And so if you take the cyclic subgroup generated by G and you take the order of that, that gives you the order of the cyclic subgroup generated by G. We also call that the order of the element itself. So the order, and so we'll denote this as the order of G here, the order of little G, which we denote the same way, this is going to be the, this will be the order of the cyclic subgroup generated by G, which an equivalent way of defining that is that the order of little G will be the smallest positive integer such that G to the n is equal to the identity of the group. Now if no such integer exists, we say that G has infinite order and that's because in that case G would produce something isomorphic to the infinite cyclic group. Now let's take an example, Z6, look at orders of elements there. Well we claim the identity element zero will have order one. And that idea there is if you take zero, if you take the cyclic subgroup generated by zero, you end up with just zero. There's only one element in there so you're going to get just one. And this happens in general that the order of the identity is always equal to one. In fact an element is the identity if and only if its order equals one. Now let's take the order of the number one in Z6, that is going to be six itself. So remember that the cyclic subgroup generated by one, this would be every element of the group. You get zero, one, two, three, four, five, and then six again. You get all six elements there. And so notice that the smallest power, or I should say in this case the smallest multiple, you get that six times one is equal to zero, but no smaller power is going to work. Like if I did five times one, that's just five. And if you do four times one, that's just five. You never get the smallest power, in this case think of it additively, the smallest multiple of one that gives you the identity zero is going to be six itself. The order of two, mod six is going to be three. Notice that if I take one times two, I get two. If I take two times two, I get four. If I get three times two, I get six, which is zero. And so then that's the smallest multiple of two that's going to give you back the identity. And so that's three. Of course those are the same elements of the subgroup, right, zero, two, and four. This is the subgroup generated by two. Notice next that the order of three is two. The cyclic subgroup generated by three is just zero and three itself. If you take the next step, you'll get back to the identity. And the order of four is itself three. Notice the cyclic subgroup generated by four is actually identical to the cyclic subgroup generated by two. And these contain three elements. That's the order of the element. And then lastly, the order of five would be six. Just like one up there, the cyclic subgroup generated by five is equal to the cyclic subgroup generated by one. And these both contain six elements. One and five are inverses, so they generate the same cyclic subgroup. And hence the orders will always be the same. Now the order does depend on the group in play. If we think of, if we want to ask ourselves, what are the orders when we do multiplication for these operations here? Well the idea is, well Z, Z six star. This only contains two elements, one and five. And one is the identity, so its order will be one. And then the order of five will actually be two because the cyclic subgroup generated by five is actually all of Z six star. You get the identity, you get five, and then you get five times five, which is 25, which reduces mod six to be one again. So the order does depend on the group itself. You'll see that one and five have orders one and two when you think of multiplication, but they have order six when you consider the addition operation there.