In which we prove Lagrange’s theorem, and deduce many interesting results as a consequence.
- Lemma 33 (Coset equality test): Let be a subgroup of a group . Take , . Have if and only if . For one direction we noted that so if the two left cosets are the same then and that gave us . For the other direction, we showed that and similarly the other way round.
- Theorem 34 (Lagrange‘s Theorem): Let be a finite group and let be a subgroup of . Then . We showed that the left cosets of partition , and that each left coset of has the same size as . These two immediately give .
- Lemma 35: Let be a finite group. Take . Then $g$ has finite order. We noted that , , , … all lie in the finite set , so we must have some repetition , from which the result followed.
- Corollary 36: Let be a finite group. Take . Then divides . We noted that is a subgroup of with order .
- Corollary 37: Let be prime, let be a finite group with order . Then is cyclic. If , then we saw that , so is a generator of .
- Corollary 38: Let be a finite group, take . Then . This follows from the facts that and .
- Theorem 39 (Fermat‘s Little Theorem): Let be prime. Let be an integer coprime to . Then . This is immediate from Corollary 38, since is a group of order .
- Theorem 40 (Fermat-Euler Theorem): Let be an integer. Define . Let be an integer coprime to . Then . This is another consequence of Corollary 38, this time noting that is a group of order .
Understanding today’s lecture
Can you relate Lemma 33 to the two examples of left cosets we saw at the end of the last lecture? Check that it works out.
Having proved Lagrange’s Theorem, I mentioned that there is an alternative way to prove that the left cosets of partition using equivalence relations. Define a relation on by if and only if . Can you show that this is an equivalence relation? What are the equivalence classes? Now complete the proof.
Why is Fermat’s Little Theorem a special case of the Fermat-Euler Theorem?
When I reminded you of what an equivalence relation is, I gave as an example the relation on a group , defined by if and only if or . Can you show that this really is an equivalence relation? What are the equivalence classes?
Let be a prime. Which elements of (a group under multiplication) are self-inverse (have )? What are the equivalence classes for the equivalence relation from the last paragraph when applied to this group? What does this tell us about the product modulo ? The result here is called Wilson‘s theorem; you can read more about it in Richard Earl’s online notes, or in my blog post.
Let be a finite group with even order. Can you show that must contain an element of order 2? (Hint: use the equivalence relation from the paragraph before last.) Again, you can read more about this in the online notes, but I strongly recommend that you try to deduce it for yourself first, it’s good practice.
I’ve written about several of these topics before on this blog. I’ve written about Lagrange’s Theorem (not be confused with another Lagrange’s Theorem that I’ve also written about), and about Fermat’s Little Theorem (that post outlines the three proofs I mentioned in today’s lecture). You’ll find a handy video with our statement and proof of Lagrange’s Theorem at hedgehogmaths. Remember to watch out for tigers.
The Euler totient function has many interesting properties. Can you show that if is prime then ? What is for a general integer ? It turns out that is multiplicative: if and are coprime, then . (Note that it’s important here that and are coprime!) Can you prove this? There’s a nice connection with the Chinese Remainder Theorem. You can read more about the function in a number theory book (such as The Higher Arithmetic by Davenport), or online (eg of course Wikipedia has a page).
Here’s some information from Wikipedia about the RSA cryptosystem that I mentioned at the end of the lecture as using Fermat-Euler.
Preparation for Lecture 9
Lecture 9 is weeks away, so for now I suggest that you prepare by working on the material from the first four weeks of the course. If you understand that really well, then you’ll be ready for next term. I’ll put some more specific questions up for you to consider a few days before Lecture 9, so check back nearer the time. Here’s one for you to ponder for now.
Let and be isomorphic groups, and let be an isomorphism. Take . What can you say about the orders of in and in ? How are they related?