Week 4: Lagrange’s Theorem

Welcome to our activities for Week 4 (the week of September 26). Our only theme this week is Lagrange’s Theorem, which might be considered the fundamental theorem of finite groups. Here it is.

Theorem (Lagrange’s Theorem) Let G be a finite group, and let H be a subgroup of G. Then the number of elements in H is a divisor of the number of elements of G.

Remarks on Lagrange’s Theorem

• Recall that the number of elements in a group is called the order of the group. So Lagrange’s Theorem says that the order of a subgroup of a finite group divides the order of the group. (Recall that “divides” means “is a factor of” when we say that one integer divides another).
• Lagrange’s Theorem says, for example, that there is no point in looking for a subgroup with eight elements in a group with 30 elements; none can exist because 8 is not a divisor of 30.
• Another consequence of Lagrange’s Theorem is that no proper subgroup of a finite group can contain more than half of the group’s elements.

Our goal this week is to prove Lagrange’s Theorem. That it is possible to do so with nothing at our disposal except the axioms of a group is remarkable. It is also (in my opinion) a compelling piece of evidence of the power of the abstract axiomatic approach to algebra. By reasoning from the axioms defining a group, we will be able to deduce the statement of the theorem, which then specializes to all sorts of contexts such as permutations, symmetries, finite groups of numbers, etc. The proof uses the full content of the definition of a group (associativity of the operation, existence of an identity element and inverses for every element). A good exercise when you are studying the proof, is to look our for exactly where each of these ingredients is used in the reasoning. There is one technical concept that we have not met yet, which we need for this proof. That is the idea of the left cosets determined by an element and a subgroup of a group. As usual we have two short video presentations on this week’s content. The first one describes what is meant by a coset and explains the mechanism of the proof of Lagrange’s Theorem, which relies on a key property of cosets. The second video proves this key property of cosets. You can watch them in either order, both are needed to complete the proof of the Theorem. Full details are in Section 2.1 of the lecture notes.

Activity for Week 4 consists of the following steps.

1. Make sure you are familiar with the content of Chapter 1 of the lecture notes. This completes the “first quarter” of the module, which is the relevant content for the first of our four exam questions.
2. Come to the lectures on Thursday (on Lagrange’s Theorem and its significance) and Friday (on how to prove it using properties of cosets).
   Slides for this week’s lectures
   If you cannot make it to the lectures, have a look at these video lectures on the same topics (as usual, please ignore any details that are particular to another year).
   
   
   
   
3. Review the lecture notes for Section 2.1, on Lagrange’s Theorem.
   
4. Continue to work on the first of our two homework assignments, which is currently open at the “Homework” section of the MA3343 Blackboard page, with a deadline of Friday October 14. Submission is via the “Homework” section of the Blackboard page, as a single pdf please.
5. Tutorials commence this week, on Wednesday at 2.00 in Room IT-125G, and Thursday at 1.00 in ENG-2033. The tutor is Koushik Paul, who is a research student working in group theory. It is sufficient to participate in one tutorial per week. There will be an opportunity there to discuss the curriculum content and homework problems, and to ask questions.