Lecture 2: Subgroups and Cyclic Groups

Lecture 2: Subgroups and Cyclic Groups 2 Subgroups and Cyclic Groups 2.1 Review Last time, we discussed the concept of a group, as well as examples of groups. In particular, a group is a set G with an associative composition law G×G −→ G that has an identity as well inverses for each element with respect to the composition law ×. Our guiding example was that of the group of invertible n×n matrices, known as the general linear group (GLn (R) or GLn (C), for matrices over R and C, respectively.) Example 2.1 Let GLn (R) be the group of n×n invertible real matrices. • Associativity. Matrix multiplication is associative; that is, (AB)C = A(BC), and so when writing a product consisting of more than two matrices, it is not necessary to put in parentheses.   1 ··· 0   • Identity. The n×n identity matrix is In =  ... . . . ... , which is the matrix with 1s along the 0 ··· 1 diagonal and 0s everywhere else. It satisfes the property that AI = IA = A for all n×n matrices A. • Inverse. By the invertibility condition of GLn , every matrix A ∈ GLn (R) has an inverse matrix A−1 such that AA−1 = A−1 A = In . Furthermore, each of these matrices can be seen as a transformation from Rn −→ Rn , taking each vector ⃗v to A⃗v . That is, there is a bijective correspondence between matrices A and invertible transformations TA : Rn −→ Rn taking TA (⃗v ) = A⃗v . Another example that showed up was the integers under addition. Example 2.2 The integers Z with the composition law + form a group. Addition is associative. Also, 0 ∈ Z is the additive identity, and −a ∈ Z is the inverse of any integer a. On the other hand, the natural numbers N under addition would not form a group, because the invertibility condition would be violated. Lastly, we looked at the symmetric group Sn . Example 2.3 The symmetric group Sn is the permutation group of {1, · · · , n}. 2.2 Subgroups In fact, understanding Sn is important for group theory as a whole because any fnite group "sits inside" Sn in a certain way9 , which we will begin to discuss today. Guiding Question What does it mean for a group to "sit inside" another group? If a subset of a group satisfes certain properties, it is known as a subgroup. 9 This is known as Cayley’s Theorem and is discussed further in section 7.1 of Artin. Lecture 2: Subgroups and Cyclic Groups Defnition 2.4 Given a group (G, ·), a subset H ⊂ G is called a subgroup if it satisfes: • Closure. If h1 , h2 ∈ H, then h1 · h2 ∈ H. • Identity. The identity element e in G is contained in H. • Inverse. If h ∈ H, its inverse h−1 is also an element of H. As notation, we write H ≤ G to denote that H is a subgroup of G. Essentially, these properties consists solely of the necessary properties for H to also be a group under the same operation ·, so that it can be considered a subgroup and not just some arbitrary subset. In particular, any subgroup H will also be a group with the same operation, independent of the larger group G. Example 2.5 The integers form a subgroup of the rationals under addition: (Z, +) ⊂ (Q, +). The rationals are more complicated than the integers, and studying simpler subgroups of a certain group can help with understanding the group structure as a whole. Example 2.6 The symmetric group S3 has a three-element subgroup {e, (123), (132)} = {e, x, x2 }. However, the natural numbers N = {0, 1, 2, · · · } ⊂ (Z, +) are not a subgroup of the integers, since not every element has an inverse. Example 2.7 The matrices with determinant 1, called the special linear group, form a subgroup of invertible matrices: SLn (R) ⊂ GLn (R). The special linear group is closed under matrix multiplication because det(AB) = det(A) det(B). 2.3 Subgroups of the Integers The integers (Z, +) have particularly nice subgroups. Theorem 2.8 The subgroups of (Z, +) are {0}, Z, 2Z, · · · .a a Where n ∈ Z, nZ consists of the multiples of n, {nx : x ∈ Z}. This theorem demonstrates that the condition that a subset H of a group be a subgroup is quite strong, and requires quite a bit of structure from H. Proof. First, nZ is in fact a subgroup. • Closure. For na, nb ∈ nZ, na + nb = n(a + b). • Identity. The additive identity is in nZ because 0 = n · 0. • Inverse. For na ∈ nZ, its inverse −na = n(−a) is also in nZ. Now, suppose S ⊂ Z is a subgroup. Then clearly the identity 0 is an element of S. If there are no more elements in S, then S = {0} and the proof is complete. Otherwise, pick some nonzero h ∈ S. Without loss of generality, we assume that h > 0 (otherwise, since −h ∈ S as well by the invertibility condition, take −h instead of h.) Thus, S contains at least one positive integer; let a be the smallest positive integer in S. Then we claim that S = aZ. If a ∈ S, then a + a = 2a ∈ S by closure, which implies that 2a + a = 3a ∈ S, and so on. Similarly, −a ∈ S by inverses, and −a + (−a) = −2a ∈ S, and so on, which implies that aZ ⊂ S. Lecture 2: Subgroups and Cyclic Groups Now, take any n ∈ S. By the Euclidean algorithm, n = aq + r for some 0 ≤ r < a. From the subgroup properties, n − aq = r ∈ S as well. Since a is the smallest positive integer in S, if r > 0, there would be a contradiction, so r = 0. Thus, n = aq, which is an element of aZ. Therefore, S ⊂ aZ. From these two inclusions, S = aZ and the proof is complete. Corollary 2.9 Given a, b ∈ Z, consider S = {ai + bj : i, j ∈ Z}. The subset S satisfes all the subgroup conditions, so by Theorem 2.8, there is some d such that S = dZ. In fact, d = gcd(a, b). Proof. Let e = gcd(a, b). Since a ∈ S, a = dk and b = dℓ for some k, ℓ. Since the d from before divides a and b, it must also divide e, by defnition of the greatest common divisor. Also, since d ∈ S, by the defnition of S, d = ar + bs for some r and b. Since e divides a and b, e divides both ar and bs and therefore d. Thus, d divides e, and e divides d, implying that e = d. So S = gcd(a, b)Z. In particular, we have showed that gcd(a, b) can always be written in the form ar + bs for some r, s. 2.4 Cyclic Groups Now, let’s discuss a very important type of subgroup that connects back to the work we did with (Z, +). Defnition 2.10 Let G be a group, and take g ∈ G. Let the cyclic subgroup generated by g be ⟨g⟩ := a {· · · g −2 , g −1 , g 0 = e, g 1 , g 2 , · · · } ≤ G. a The := symbol is usually used by mathematicians to mean "is defned to be." Other people may use ≡ for the same purpose. Since g a · g b = g a+b , the exponents of the elements of a cyclic subgroup will have a related group structure to (Z, +). Example 2.11 The identity element generates the trivial subgroup {e} = ⟨e⟩ of any group G. There are also nontrivial cyclic subgroups. Example 2.12 In S3 , ⟨(123)⟩ = {e, (123), (132)}. Evidently, a cyclic subgroup of any fnite group must also be fnite. Example 2.13 Let C × be the group of nonzero complex numbers under multiplication. Then 2 ∈ C will generate ⟨2⟩ = {· · · , 1/4, 1/2, 1, 2, 4, · · · .} On the other hand, i ∈ C will generate ⟨i⟩ = {1, i, −1, −i}. This example shows that a cyclic subgroup of an infnite group can be either infnite or fnite.10 10 Can you work out the cases for which g ∈ C the cyclic subgroup of C × is fnite or infnite? Lecture 2: Subgroups and Cyclic Groups Guiding Question What does a cyclic subgroup look like? Can they be classifed? Theorem 2.14 Let S = {n ∈ Z : g n = e}. Then S is a subgroup of Z, so S = dZ or S = {0}, leading to two cases: • If S = {0}, then ⟨g⟩ is infnite and all the g k are distinct. • If S = dZ, then ⟨g⟩ = {e, g, g 2 , · · · , g d−1 } ⊂ G, which is fnite. Proof. First, S must be shown to actually be a subgroup of Z. • Identity. The identity 0 ∈ S because g 0 = e. • Closure. If a, b ∈ S, then g a = g b = e, so g a+b = g a g b = e · e = e, so a + b ∈ S. • Inverse. If a ∈ S, then g −a = (g a )−1 = e−1 = e, so a ∈ S. Now, consider the frst case. If g a = g b for any a, b, then multiplying on right by g −b gives g a · g −b = g a−b = e. Thus, a − b ∈ S, and if S = {0}, then a = b. So any two powers of g can only be equal if they have the same exponent, and thus all the g i are distinct and the cyclic group is infnite. Consider the second case where S = dZ. Given any n ∈ Z, n = dq + r for 0 ≤ r < d by the Euclidean algorithm. Then g n = g dq · g r = g r , which is in {e, g, g 2 , · · · , g d−1 }. Defnition 2.15 So if d = 0, then ⟨g⟩ is infnite; we say that g has infnite order. Otherwise, if d ̸= 0, then |⟨g⟩| = d and g has order d. It is also possible to consider more than one element g. Defnition 2.16 Given a subset T ⊂ G, the subgroup generated by T is ⟨T ⟩ := {te11 · · · tenn | ti ∈ T, ei ∈ Z}. Essentially, ⟨T ⟩ consists of all the possible products of elements in T. For example, if T = {t, n}, then ⟨T ⟩ = {· · · , t2 n−3 t4 , n5 t−1 , · · · }. Defnition 2.17 If ⟨T ⟩ = G, then T generates G.a a Given a group G, what is the smallest set that generates it? Try thinking about this with some of the examples we’ve seen in class! Example 2.18 The set {(123), (12)} generates S3 . Example 2.19 The invertible matrices GLn (R) are generated by elementary matricesa . a The matrices giving row-reduction operations.