 So let's go back to the multiplicative group. If n is prime, then the elements 1, 2, 3, up to n minus 1 form a group under multiplication. Now, if a is less than n, then Euler's result proved that there is a least k for which a to power k is congruent to one mod n. Moreover, we know that the powers of a are closed under multiplication, and every power of a has an inverse. That's another power of a. And so our set 1, a, a squared, and so on up to a to power k minus 1 also forms a group. Now, this group is formed by the powers of a single element, and it's a common enough thing to do that we have a specific name for it. We call this the cyclic group generated by a, and the notation we use is something like bracket a. Now, since the elements of this cyclic group are part of an existing group, we also call it a subgroup. And formally, we'll define a subgroup as follows. Let g star be a group. If h is a subset of g, and h star is also a group, then we say that h is a subgroup of g. So, for example, we might find a subgroup of the multiplicative group of integers mod 11. And so we know that the group itself consists of the integers from 1 through 10 under multiplication mod 11. And we can form a subgroup by finding the powers of any element. So, let's pick, how about 4? Suppose we find the powers of 4. So, we choose 4 and find the powers of 4 mod 11. So 4 to the second, that's 16, which reduces to 5. 4 to the third, again, one of the advantages to working mod n is we never have to work with large numbers. 4 to the third is the same as 4 to the second times 4. And we know the 4 to the second is congruent to 5. And so 4 to the third is congruent to 9. 4 to the fourth, well, that's 4 to the third times 4. And we already know what 4 to the third is. And so this is 3. 4 to the fifth is 1. And we're back where we started from. And so the subgroup generated by 4 consists of the elements 1, 3, 4, 5, and 9. And so now we'll introduce two definitions. Unfortunately, mathematicians are terrible at coming up with new names, and so our two definitions are actually for the same word, but we'll have somewhat different meanings. So the first definition, let G be a group. The order of the group is the cardinality of G. That is to say the number of elements in the group. The other definition we'll introduce, let G be a group with an identity E. The order of an element is the least k for which a to the power k is equal to the identity. So order has two different meanings. It could either refer to the number of elements in the group, or it could refer to the power that gives you the identity for the first time. So going back to our multiplicative group of integers, mod 11, we'll find the order and the order of 4. And so we know that our group itself had 10 elements. So the order is 10. And we also saw the first time we got the identity was 4 to power 5. And so the element 4 has order 5. And if you look carefully, you can see why even though the term has two different definitions, they are pretty closely related. The fact that 4 has order 5 also reflects in the fact that the subgroup that we get has 5 elements. So where does that put us? Well, we found that if G is the multiplicative group of integers mod n, n is prime, then we can say that the order of G is n minus 1. There's n minus 1 elements in the group. And the order of an element of G is a divisor of G minus 1. And since G is a group and any element generates a subgroup, this suggests a more general theorem where we can take any group and any subgroup generated by an element. So the order of a subgroup generated by an element is a divisor of the order of the group. Well, we are claiming this is a theorem, so we should have a proof for it. Fortunately, Euler's proof for the integers mod n under multiplication can be re-tasked for the proof of this theorem. And again, I'll leave that as a homework problem.