Lagrange s Theorem. Philippe B. Laval. Current Semester KSU. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10

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1 Lagrange s Theorem Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 1 / 10

2 Introduction In this chapter, we develop new tools which will allow us to extend to every finite group some of the results we already know for cyclic groups. More specifically, we will be able to generalize results regarding the order of a subgroup. We know that in the case of cyclic groups, the order of every element and hence of every subgroup must divide the order of the group. The tools we develop here will allow us to generalize this result to every finite group. The above statement comes from this result we saw when we were studying cyclic groups: if G = a and b G then ord (b) is a factor of ord (a) that is ord (b) divides ord (a) which is the same as saying ord (a) = k ord (b) where k is some strictly positive integer. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 2 / 10

3 Lagrange s Theorem We now state and prove the main theorem of these slides: Lagrange s theorem. Though, as we will see, the proof is not diffi cult as we did most of the work in the previous slides on cosets. Joseph-Louis Lagrange ( ) was a French mathematician born in Italy. He made significant contributions to the fields of analysis, number theory, and both classical and celestial mechanics. Theorem (Lagrange s theorem) Let G be a finite group and H a subgroup of G. H divides G in other words, the order of H divides the order of G. Moreover, the number of distinct left (right) cosets of H is G H. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 3 / 10

4 Lagrange s Theorem: Sketch of a Proof We discuss the proof for left cosets. The proof for right cosets is similar. Suppose that G has n distinct left cosets we will call a i H, i = 1, 2,..., n. What can be said about these distinct left cosets with respect to G? Write G in term of these cosets. Using the above, write G in terms of H and n. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 4 / 10

5 Lagrange s Theorem Remark: Let us make a few remarks regarding Lagrange s theorem. 1 The theorem provides a subgroup candidate test. It says that a group can only have subgroups of order which divide the order of the group. For example, a group of order 10 can only have subgroups of orders 1, 2, 5 and However, and this is important, the theorem does not say there will be a subgroup that has the order of every divisor of a group. So, if a group of order 10 can only have subgroups of orders 1, 2, 5 and 10, it does not mean there will be subgroups for each of these orders. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 5 / 10

6 Lagrange s Theorem Definition The index of a subgroup H in a group G, denoted [G : H] is the number of distinct left (or right) cosets of H in G. Corollary Let G be a finite group and H a subgroup of G. [G : H] = G H. Proof. See Lagrange s theorem. Example Let G = (Z 12, +) and H = 4. What is H and [G : H]. Find all the left cosets of H. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 6 / 10

7 Consequences of Lagrange s Theorem We look at some consequences of this important theorem. Some are related to the order of a group, some are not. Theorem Let G be a finite group and a G. Then, ord (a) divides G and consequently, a G = e. Theorem Let G be a group such that G = p and p is prime. Then, G must be cyclic. Moreover, any element of G other than e is a generator for G. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 7 / 10

8 Consequences of Lagrange s Theorem Sketch of the proof of theorem 1: Express ord (a) in term of the order of a subgroup of G involving a. Using Lagrange s theorem, express G in terms of ord (a). Using the above, compute a G. Sketch of the proof of theorem 2: If p is prime, then p > 1. Let x G, what can be said about x? Conclude since x was arbitrary. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 8 / 10

9 Consequences of Lagrange s Theorem Example What are the possible non-trivial subgroups of (Z 7, +)? Example Same question for (Z 12, +). Example Use the above two theorems to show that every group of order 5 is Abelian. Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 9 / 10

10 Exercises Do the following problems: 1 Let p and q be two distinct prime numbers and suppose G is a group of order pq. Show that every proper subgroup of G is cyclic. Recall a proper subgroup of G is a subgroup of G which is not equal to G. 2 Let G be a group and H a subgroup of G. Let ah be a left coset of H other than H. Prove ah is not a subgroup. 3 Let G be a group of order p 2 where p is prime. Show G must have a subgroup of order p. 4 Do the following problems from chapter 13 in your book: B2, C1, C2, C4, Philippe B. Laval (KSU) Lagrange s Theorem Current Semester 10 / 10