ABSTRACT ALGEBRA 1, LECTURES NOTES 4: THE QUATERNION GROUP, LINEAR GROUPS, WHY ALL THESE EXAMPLES?

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1 ABSTRACT ALGEBRA 1, LECTURES NOTES 4: THE QUATERNION GROUP, LINEAR GROUPS, WHY ALL THESE EXAMPLES? ANDREW SALCH 1. The quaternion group. The quaternion group, written Q 8, is another example (other than D 8 ) of a nonabelian group of order 8; it shows up often enough that it gets its own name. Definition 1.1. The quaternion group is the group with elements {1, 1, i, i, j, j, k, k}, and multiplication ( 1) 2 = 1, ( 1) x = x for all x, 1 = i 2 = j 2 = k 2, i j = k, jk = i, ki = j, ji = k, k j = i, ik = j. Proposition 1.2. The quaternion group Q 8 has the presentation i, j i 4 = 1, i 2 = j 2, ji = i j 1. Proof. Clearly the relations in the group i, j i 4 = 1, i 2 = j 2, ji = i j 1 are also satisfied in Q 8 (this is easier if you notice that i 1 = i and j 2 = j in Q 8 ), and the elements i and j generate Q 8 as well as i, j i 4 = 1, i 2 = j 2, i 1 ji = j 1. So all that s left is to check that Q 8 doesn t have any more relations beyond i 4 = 1, i 2 = j 2, and ji = i j 1. So all we need is to check that Q 8 and i, j i 4 = 1, i 2 = j 2, ji = i j 1 have the same number of elements. If we write the elements of i, j i 4 = 1, i 2 = j 2, ji = i j 1 with all the powers of i to the left of the powers of j (using the relation ji = i j 1 ), we get the list of elements {1, i, i 2, i 3, j, i j, i 2 j, i 3 j} of i, j i 4 = 1, i 2 = j 2, ji = i j 1, and the relations i 4 = 1, i 2 = j 2, and ji = i j 1 tells us that this a complete list of the elements of i, j i 4 = 1, i 2 = j 2, ji = i j 1. That s eight elements, the same as Q 8. Remark 1.3. Be careful that you don t get too sloppy when making these kinds of arguments where, after showing that two finite groups have the same generating set and the relations in one group also hold in the other, if the two groups have the same order then they are isomorphic; you really need those hypotheses that I put in italics to hold before Date: September

2 2 ANDREW SALCH you can conclude that two groups of the same order are isomorphic! You already know some examples of two groups of the same order that are not isomorphic: you know that Z/6Z can t be isomorphic to Σ 3, even though they are both of order six, since Z/6Z is abelian and Σ 3 is not abelian. You also showed, in an earlier homework exercise, that D 24 is not isomorphic to Σ 4, even though they are both non-abelian groups of order 24. In the next homework exercise, you will show that there even exist two nonabelian groups of the same order and which can be generated by the same number of generators (namely, two generators), which are non-isomorphic. Exercise 1.4. Prove that the group Q 8 is not isomorphic to the group D Linear groups. Definition 2.1. Let R be either the integers Z, the rational numbers Q, the real numbers R, the complex numbers C, or the mod n integers Z/nZ for some positive integer n. By the nth general linear group over R, written GL n (R), we mean the group of invertible n-by-n matrices with entries in R, with matrix multiplication as the group operation. By the nth special linear group over R, written S L n (R), we mean the group of n- by-n matrices with entries in R which have determinant equal to 1, with matrix multiplication as the group operation. In Definition 2.1, there is really no need for R to be one of the possibilities listed (Z, Q, R, C, Z/nZ): all you need in order to do matrix multiplication and to speak of invertible matrices and determinants of matrices is for R to be a commutative ring, of which Z, Q, R, C, and Z/nZ are only special cases. Some of you have seen commutative rings before, and some haven t, but we will get to them soon enough. Exercise 2.2. Compute presentations for the groups GL 2 (Z/2Z) and S L 2 (Z/3Z). The linear groups are important largely because they import a huge amount of linear algebra into group theory: basically every theorem about invertible square matrices is a theorem about the general linear groups, and if there is some group G we would like to understand, we can sometimes get the information we need about G by finding a way to describe G as a subgroup of GL n (R) for some R and some n (in other words, by finding a one-to-one group homomorphism G GL n (R)), and using linear algebra to deduce various properties of G. An example of this interplay of linear algebra and group theory is in Exercise 2.5, but first we need a preliminary definition: Definition 2.3. Let G be a group, and let H, H be subgroups of G. We say that H is conjugate to H if there exists an element g G such that g 1 Hg = H, that is, for each element h H, the element g 1 hg is in H, and for each element h H, the element g 1 h g is in H. Example 2.4. Remember the presentation Σ 3 = σ, τ σ 2 = 1, τ 3 = 1, τ 2 σ = σ τ for the symmetric group on three symbols. Here is an easy computation: τ 1 στ = τ 2 στ = τ 2 τ 2 σ = τσ,

3 ABSTRACT ALGEBRA 1, LECTURES NOTES 4: THE QUATERNION GROUP, LINEAR GROUPS, WHY ALL THESE EXAMPLES?3 so the subgroup {1, σ} of Σ 3 is conjugate to the subgroup {1, τσ}. We say that two subgroups of G are in the same conjugacy class if one subgroup is conjugate to the other. Exercise 2.5. Let n be a positive integer. Compute the conjugacy classes of subgroups of order 2 in GL n (C). (Hint: this is hard unless you think back on some results you learned in linear algebra, and then it s rather easy. In particular, review the Jordan canonical form of a square matrix over an algebraically closed field.) 3. Why all these finite groups? As we work through chapters 3, 4, and 5 of your textbook, we will be studying powerful tools which allow you to classify finite groups; for example, given a finite group G, we will have systematic ways of breaking G into smaller pieces which can be studied more easily, given two finite groups of the same order, we will have systematic ways of checking whether they are isomorphic, and, given a positive integer n, we will have ways of finding the number of isomorphism classes of groups of order n. One reason we have spent the past week studying all these examples of finite groups symmetric groups, cyclic groups, dihedral groups, quaternion groups, linear groups is that these groups are simple and ubiquitous enough that they tend to show up as building blocks in other groups, and it will be very helpful to be able to recognize these groups when they arise as subgroups or quotient groups of other, more exotic groups. (These families of groups are also each important in their own right and are heavily used and studied.) It s reasonable now to ask just how many finite groups there are of a given order is every finite groups symmetric or cyclic or dihedral or quaternion, etc.? The answer is no. First, here is an easy exercise which establishes a basic fact about the number of groups of order n that it is finite: Exercise 3.1. Let n be a positive integer. Prove that the number of isomorphism classes of groups of order n is finite. So, for each positive integer n, the number of isomorphism classes of order n is some positive integer. Let s call this integer f (n). By the end of the semester you will be computing f (n) yourself, for many values of n. For now, here is a table of f (n) for small n, to give you a sense of roughly how many groups there are of a given order:

4 4 ANDREW SALCH n f(n)

5 ABSTRACT ALGEBRA 1, LECTURES NOTES 4: THE QUATERNION GROUP, LINEAR GROUPS, WHY ALL THESE EXAMPLES?5 n f(n)