Lecture Notes on Subgroups

2. Subgroups Consider the chain of inclusions of groups Z ⊂ Q ⊂ R ⊂ C. where the law of multiplication is ordinary addition. Then each subset is a group, and the group laws are obviously com­ patible. That is to say that if you want to add two integers together, it does not matter whether you consider them as integers, rational num­ bers, real numbers or complex numbers, the sum always comes out the same. Definition 2.1. Let G be a group and let H be a subset of G. We say that H is a subgroup of G, if the restriction to H of the rule of multiplication and inverse makes H into a group. Notice that this definition hides a subtlety. More often than not, the restriction to H × H of m, the rule of multiplication of G, won’t even define a rule of multiplication on H itself, because there is no a priori reason for the product of two elements of H to be an element of H. For example suppose that G is the set of integers under addition, and H is the set of odd numbers. Then if you take two elements of H and add them, then you never get an element of H, since you will always get an even number. Similarly, the inverse of H need not be an element of H. For example take H to be the set of natural numbers. Then H is closed under addition (the sum of two positive numbers is positive) but the inverse of every element of H does not lie in H. Definition 2.2. Let G be a group and let S be subset of G. We say that S is closed under multiplication, if whenever a and b are in S, then the product of a and b is in S. We say that S is closed under taking inverses, if whenever a is in S, then the inverse of a is in S. For example, the set of even integers is closed under addition and taking inverses. The set of odd integers is not closed under addition (in a big way as it were) and it is closed under inverses. The natural numbers are closed under addition, but not under inverses. Proposition 2.3. Let H be a non-empty subset of G. Then H is a subgroup of G iff H is closed under multiplication and taking inverses. Furthermore, the identity element of H is the identity element of G and the inverse of an element of H is equal to the inverse element in G. If G is abelian then so is H. 1 MIT OCW: 18.703 Modern Algebra Prof. James McKernan Proof. If H is a subgroup of G, then H is closed under multiplication and taking inverses by definition. So suppose that H is closed under multiplication and taking inverses. Then there is a well defined product on H. We check the axioms of a group for this product. Associativity holds for free. Indeed to say that the multiplication on H is associative, is to say that for all g, h and k ∈ H, we have (gh)k = g(hk). But g, h and k are elements of G and the associative rule holds in G. Hence equality holds above and multiplication is associative in H. We have to show that H contains an identity element. As H is non-empty we may pick a ∈ H. As H is closed under taking inverses, a−1 ∈ H. But then e = aa−1 ∈ H as H is closed under multiplication. So e ∈ H. Clearly e acts as an identity element in H as it is an identity element in G. Suppose that h ∈ H. Then h−1 ∈ H, as H is closed under taking inverses. But h−1 is clearly the inverse of h in H as it is the inverse in G. Finally if G is abelian then H is abelian. The proof follows just like the proof of associativity. D Example 2.4. (1) The set of even integers is a subgroup of the set of integers under addition. By (2.3) it suffices to show that the even integers are closed under addition and taking inverses, which is clear. (2) The set of natural numbers is not a subgroup of the group of integers under addition. The natural numbers are not closed under taking inverses. (3) The set of rotations of a regular n-gon is a subgroup of the group Dn of symmetries of a regular n-gon. By (2.3) it suffices to check that the set of rotations is closed under multiplication and inverse. Both of these are obvious. For example, suppose that R1 and R2 are two rotations, one through θ radians and the other through φ. Then the product is a rotation through θ + φ. On the other hand the inverse of R1 is rotation through 2π − θ. (4) The group Dn of symmetries of a regular n-gon is a subgroup of the set of invertible two by two matrices, with entries in R. Indeed any symmetry can be interpreted as a matrix. Since we have already seen that the set of symmetries is a group, it is in fact a subgroup. 2 MIT OCW: 18.703 Modern Algebra Prof. James McKernan (5) The following subsets are subgroups. Mm,n (Z) ⊂ Mm,n (Q) ⊂ Mm,n (R) ⊂ Mm,n (C). (6) The following subsets are subgroups. GLn (Q) ⊂ GLn (R) ⊂ GLn (C). (7) It is interesting to enumerate the subgroups of D3 . At one ex­ treme we have D3 and at the other extreme we have {I}. Clearly the set of rotations is a subgroup, {I, R, R2 }. On the other hand {I, Fi } forms a subgroup as well, since Fi2 = I. Are these the only subgroups? Suppose that H is a subgroup that contains R. Then H must contain R2 and I, since H must contain all powers of R. Sim­ ilarly if H contains R2 , it must contain R4 = (R2 )2 . But R4 = R3 R = R. Suppose that in addition H contains a flip. By symmetry, we may suppose that this flip is F = F1 . But RF1 = F3 and F R = F2 . So then H would be equal to G. The final possibility is that H contains two flips, say F1 and F2 . Now F1 R = F2 , so R = F1−1 F2 = F1 F2 . So if H contains F1 and F2 then it is forced to contain R. In this case H = G as before. Here are some examples, which are less non-trivial. Definition-Lemma 2.5. Let G be a group and let g ∈ G be an element of G. The centraliser of g ∈ G is defined to be Cg = { h ∈ G | hg = gh }. Then Cg is a subgroup of G. Proof. By (2.3) it suffices to prove that Cg is closed under multiplication and taking inverses. 3 MIT OCW: 18.703 Modern Algebra Prof. James McKernan Suppoe that h and k are two elements of Cg . We show that the product hk is an element of Cg . We have to prove that (hk)g = g(hk). (hk)g = h(kg) by associativity = h(gk) as k ∈ Cg = (hg)k by associativity = (gh)k as h ∈ Cg = g(hk) by associativity. Thus hk ∈ Cg and Cg is closed under multiplication. Now suppose that h ∈ G. We show that the inverse of h is in G. We have to show that h−1 g = gh−1 . Suppose we start with the equality hg = gh. Multiply both sides by h−1 on the left. We get h−1 (hg) = h−1 (gh), so that simplifying we get g = (h−1 g)h. Now multiply both sides of this equality by h−1 on the right, gh−1 = (h−1 g)(hh−1 ). Simplifying we get ghg −1 = g −1 h, which is what we want. Thus h−1 ∈ Cg . Thus Cg is closed under taking D inverses and Cg is a subgroup by (2.3). Lemma 2.6. Let G be a finite group and let H be a non-empty finite set, closed under multiplication. Then H is a subgroup of G. Proof. It suffices to prove that H is closed under taking inverses. Let a ∈ H. If a = e then a−1 = e and this is obviously in H. So we may assume that a = e. Consider the powers of a, a, a2 , a3 , . . . . As H is closed under products, it is obviously closed under powers (by an easy induction argument). As H is finite and this is an infinite sequence, we must get some repetitions, and so for some m and n distinct positive integers am = a n . 4 MIT OCW: 18.703 Modern Algebra Prof. James McKernan Possibly switching m and n, we may assume m < n. Multiplying both sides by the inverse a−m of am , we get an−m = e. As a 6= e, n−m 6= 1. Set k = n−m−1. Then k > 0 and b = ak ∈ H. But ba = ak a = an−m−1 a = an−m = e. Similarly ab = e. Thus b is the inverse of a. Thus H is closed under taking inverses and so H is a subgroup of G by (2.3). D 5 MIT OCW: 18.703 Modern Algebra Prof. James McKernan