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GROUP THEORY 1. Groups A set G is called a group if there is a multiplication (a group operation) G×G → G, (g, h) 7→ gh such that (1) (∀f, g, h ∈ G) f (gh) = (f g)h, (∃ e ∈ G)(∀g ∈ G) eg = ge = g, (2) (∀g ∈ G)(∃ g −1 ∈ G) gg −1 = g −1 g = e. (3) The condition (1) is called the associativity. The element e in (2) is called the identity element, or the neutral element. It is easy to prove that such an element is unique. The identity element is frequently denoted simply by 1. The element g −1 in (3) is called the inverse element. For any g ∈ G, the element g −1 is unique. Note that the additive notation is used frequently instead of the multiplicative notation introduced above. In the additive notation, the group operation is denoted by +, the neutral element is denoted by 0 and is called zero, the inverse element of g is denoted by −g and is said to be the opposite element. The trivial group consisting of 1 (or 0) only is denoted by 0. A group G is called abelian (or commutative) if xy = yx for all x, y ∈ G. Usually the additive notation is used for abelian groups. In a non-abelian group, we say that two elements x and y commute if xy = yx. Let G and H be groups. A map f : G → H is called a homomorphism if f (gh) = f (g)f (h) for all g, h ∈ G. One can easily verify that f (1) = 1 and f (g −1 ) = f (g)−1 . A homomorphism f is an isomorphism (resp., epimorphism, monomorphism) if f is bijective (resp., surjective, injective). An isomorphism of G with itself is called an automorphism of G. Take an element g0 ∈ G. The map g 7→ g0−1 gg0 is an automorphism of G, which is called an inner automorphism, or a conjugation. Elements g and g0−1 gg0 are called conjugate elements. Subgroups H and g0−1 Hg0 are called conjugate subgroups. A subset H ⊆ G is called a subgroup, if H is a group with respect to the multiplication inherited from G. A subset H ⊆ G is a subgroup iff gh−1 ∈ H for all g, h ∈ H. For any homomorphism f : G → F , define the kernel Ker(f ) = f −1 (1) and the image Im(f ) = f (G). The kernel Ker(f ) is a subgroup of G and the image Im(f ) is a subgroup of F . Let H be a subgroup of G. The set aH = {ah| h ∈ H} is called a (left) coset of G with respect to H. The left cosets are the equivalence classes by the equivalence relation x−1 y ∈ H. Therefore, cosets are disjoint. The set of all left cosets is denoted by G/H. A right coset is a set of the form Ha, a ∈ G. The set of all right cosets is denoted by H\G. The number of all left cosets of G with respect to H is called 1 the index of H and is denoted by (G : H). The number of all right cosets coincides with (G : H). We say that a group G is generated by a set X ⊂ G if G is the only subgroup containing X. A group is called finitely generated if it is generated by some finite subset. Problems and theorems. 1.1. The sets Z, Q, R, C are groups with respect to addition, and the sets Q∗ = Q − 0, R∗ = R − 0, C∗ = C − 0 are groups with respect to multiplication. 1.2. Let X be an arbitrary set. The set of all bijections X → X is a group with respect to composition. This group is called the transformation group of X. 1.3. The set Aut(G) of all automorphisms of G is a group with respect to composition. 1.4. A homomorphism f : G → F is injective iff Ker(f ) = 0, the homomorphism f is surjective iff Im(f ) = F . 1.5. Prove that if H is a subgroup of G and F is a subgroup of H, then F is a subgroup of G and (G : F ) = (G : H)(H : F ). In particular, (G : H)|H| = |G|. Here |H| denotes the order of H, i.e. the number of elements in H. 1.6. A group G is said to be cyclic if there is a an element g ∈ G s.t. G consists of powers of g. Prove that any subgroup of Z is cyclic. Moreover, any subgroup of a cyclic group is cyclic. If G is cyclic and f : G → H is an epimorphism, then H is also cyclic. 1.7. The Lagrange theorem. Let G be a group and g ∈ G. Denote by n the smallest positive integer s.t. g n = 1 (if there is no such number, set n = ∞). The number n is called the order of g. The order of an element g ∈ G divides the order |G| of the group G. 1.8. For arbitrary subsets H, F ⊆ G define F H = {f h ∈ G| f ∈ F, h ∈ H}. Give an example of two subgroups H and F such that F H is not a subgroup. Prove that F H is a subgroup if and only if F H = HF . 1.9. Suppose that xn = 1 for any x ∈ G. Then |G| divides a power of n. 1.10. Let p be a prime number. Prove that any group of order p is cyclic. 1.11. If two cyclic groups have the same number of elements, then they are isomorphic. We denote by Cn a cyclic group with n elements (it is unique up to an isomorphism). 2. Normal subgroups and quotient groups A subgroup H ⊆ G is said to be normal (notation: H C G) if gHg −1 ⊆ H for all g ∈ G (it follows that gHg −1 = H for all g). The intersection of any family of normal subgroups is also a normal subgroup. The kernel of any homomorphism is a normal subgroup. If H C G, then G/H is endowed with a natural group structure. This group is called the quotient group. The map π : G → G/H taking each element to its coset is surjective and H = Ker π. The map π is called the canonical projection. Many questions about a group G can be reduced to a normal subgroup H C G and the quotient group G/H. A group G is called simple if it contains no nontrivial normal subgroups. 2 An element x ∈ G is called central if xy = yx for all y ∈ G. The set ZG of all central elements is called the center of G. Let H ⊆ G be a subgroup. Define the normalizer NH of H as {g ∈ G| gHg −1 ⊆ H}. This is the smallest subgroup of G, in which H is normal. If S ⊆ G is an arbitrary subset, then define the centralizer ZS of S as {g ∈ G | (∀x ∈ S) gx = xg}. This is the smallest subgroup of G s.t. S lies in the center of it. A tower G B G1 B · · · B Gr = ∅ is called a normal series. The number r is the length of the normal series. If all the quotients Gi /Gi+1 are simple, then the normal series is called compositional. Two normal series are called isomorphic if their quotients are in a one-to-one correspondence such that corresponding quotients are isomorphic. Problems and theorems. 2.1. The homomorphism theorem. Let f : G → H be a homomorphism of groups. Then f (G) is isomorphic to G/ Ker(f ). 2.2. Let F and H be subgroups of G. Prove the following statements: • If F C H ⊆ G, then H ⊆ NF . • If F ⊆ NH , then F H is a subgroup of G s.t. H C F H. 2.3. The isomorphism theorems. Prove the following statements: • Let F C H C G. Then G/H = (G/F )/(H/F ) (here “=” means that there is a natural isomorphism between the two groups). • Let F and H be subgroups of G and H ⊆ NF . Then H/(H ∩ F ) = HF/F . • Let f : G → G0 be a homomorphism and H 0 C G0 . Then H = f −1 (H 0 ) C G and G/H = G0 /H 0 . 2.4. A normal series G0 B G1 B · · · B Gn−1 B Gn = ∅ is is said to be cyclic (resp, abelian) if all the quotient groups Gi /Gi+1 are cyclic (resp., abelian). Prove that any abelian normal series can be refined to a cyclic normal series. 2.5. Any normal series admits a refinement to a compositional series. Moreover, if normal series G and H are isomorphic, then for any refinement of G there is an isomorphic refinement of H. 2.6. The Jordan-Hölder theorem. Any two normal series for G admit isomorphic refinements. Any two compositional series are isomorphic. Hint: First find isomorphic refinements of G B G1 B · · · B Gr = ∅ and G B H1 B ∅. Set D = G1 ∩ H1 , B = G1 H1 . Note that series B B G1 B D B ∅ and B B H1 B D B ∅ are isomorphic. 2.7. Any finitely generated abelian group is a quotient of Zn . 2.8. Let G be a group. The commutator of G is defined as the subgroup G0 of G generated by {aba−1 b−1 | a, b ∈ G}. Prove that G0 C G, that G/G0 is abelian, and that for a normal subgroup N of G, the quotient G/N is abelian iff G0 ⊆ N . 2.9. Suppose that N is a normal subgroup of G and G/N is cyclic. Then G is commutative. 3. Products of groups Let G and H be groups. On the set G × H = {(g, h) | g ∈ G, h ∈ H}, introduce the group structure as follows: (g1 , h1 )(g2 , h2 ) = (g1 g2 , h1 h2 ). 3 The group G × H is called the (direct) product of G and H. Note that G × H has two distinguished normal subgroups G × {1} and {1} × H. Problems and Theorems. 3.1. The groups G × H and H × G are isomorphic. 3.2. Let H and F be two subgroups of a group G. Assume that • H ∩ F = {1}, • any element of H commutes with any element of F . Then the subgroup HF ⊆ G is isomorphic to H × F . 3.3. The Chinese Remainder Theorem. Let p and q be relatively prime positive integers. The product Cp × Cq is isomorphic to Cpq . 3.4. For a positive integer n, let ϕ(n) denote the number of positive integers from 1 to n that are relatively prime to n. Show that the group Cn has ϕ(n) generators and that |Aut(Cn )| = ϕ(n). The function ϕ is called the Euler ϕ-function. 3.5. Prove that ϕ(nm) = ϕ(n)ϕ(m) provided that m and n are relatively prime. If n = pa1 1 · · · pakk is a prime decomposition of n, then µ ¶ µ ¶ 1 1 ak −1 a1 −1 ϕ(n) = p1 (p1 − 1) · · · pk (pk − 1) = n 1 − · 1− . p1 pk 3.6. Let N be a normal subgroup of a group G. Suppose that H is a subgroup of G such that G = N H and N ∩ H = {1}. Then we say that G is a semidirect product of N and H. There exists a homomorphism G → H which is the identity on H and whose kernel is N . 3.7. Let G be an abelian group. The torsion of G is defined as the set of all elements g ∈ G such that ng = 0 for some n (we use additive notation). If G is finitely generated, then it is (isomorphic to) a direct sum of its torsion and Z k . 4. Permutations Let X be a finite set. A permutation of the set X is any one-to-one map from X to itself. It follows that this map is also onto, just by counting elements. A permutation of the set X = {x1 , . . . , xn } is also called a permutation of the elements x1 , . . . , xn . Since the nature of the elements being permuted is not important, we can usually set X = {1, 2, . . . , n}. The set of all permutations of 1, 2, . . . , n is denoted by Sn . With any permutation of a finite set, we can associate its sign. To define the sign of a permutation, it is convenient to number all elements, on which the permutation acts. However, the sign will not depend on a particular numbering. Let σ be a permutation of 1, 2, . . . , n. An inversion of σ is any pair (i, j), where 1 ≤ i < j ≤ n but σ(i) > σ(j). The sign of σ is defined as the parity of the number of inversions. I.e., the sign is 1 if the number of inversions is even and −1 if the number of inversions is odd. A permutation is said to be even or odd depending on whether its sign is equal to 1 or −1. Let i and j be two numbers from 1 to n. Consider a permutation (ij) of the numbers 1, 2, . . . , n, such that the image of i is j, the image of j is i and the image of any other element, different from i and j, is this element itself. Such permutation is called the transposition of i and j. A transposition is an odd permutation. 4 Since permutations are maps, it makes sense to talk about the composition σ ◦τ of two permutations σ and τ . This operation makes Sn a group, called a permutation group, or a symmetric group. Problems and theorems. 4.1. Any permutation is a composition of transpositions. Hint: It is always possible to reduce the number of inversions by composing a given permutation σ with a suitable transposition. Namely, there always exists an inversion (i, i+1), i.e. σ(i) > σ(i+1). Then the permutation (σ(i)σ(i+1))◦σ has one inversion less than the permutation σ. 4.2. Composing a permutation with a transposition changes the sign of the permutation. In other words, for any permutation σ and a transposition (ij), we have sign(σ ◦ (ij)) = sign((ij) ◦ σ) = −sign(σ). 4.3. The sign of the composition of two permutations is the product of their signs. 4.4. A cycle (i1 . . . ik ) is a permutation of the set {1, 2, . . . , n} such that σ(i1 ) = i2 , σ(i2 ) = i3 , ... σ(ik ) = i1 and σ(i) = i for any i different from all i1 , . . . , ik . In other terms, the elements i1 , . . . , ik are permuted in the cyclic order. The number k is called the length of the cycle. The cycle of length k has sign (−1)k−1 . 4.5. Two cycles (i1 . . . ik ) and (j1 . . . jl ) are called disjoint if the sets {i1 , . . . , ik } and {j1 , . . . , jl } are disjoint. Any permutation can be represented as a product of disjoint cycles. This representation is unique. 4.6. Suppose that σ ∈ Sn is represented as a product of disjoint cycles of lengths n1 , . . . , nk . The order of σ is the least common multiple of n1 , . . . , nk . 4.7. Let An be the set of all even permutations in Sn . Then An is a normal subgroup of Sn of index 2. The group An is called an alternating group. 4.8. The group An is generated by all cycles of length 3. Moreover, it is generated by all cycles of the form (12k) for k = 3, . . . , n. 4.9. Describe all conjugacy classes in Sn . Answer: let n = n1 + · · · + nk be a partition of n (the order of summunds is irrelevant). To each such partition, we can assign the conjugacy class Sn [n1 , . . . , nk ] consisting of all permutations that decompose into independent cycles of lengths n1 , . . . , nk . All conjugacy classes have this form. 4.10. Describe all conjugacy classes in An . 5. Group actions Let X be an arbitrary set. A transformation of X is any map from X to itself that is one-to-one and onto. All transformations of X form a group with respect to the composition. In particular, if X is finite, this is a permutation group. An action of a group G on a set X is any homomorphism from G to the transformation group of X. For g ∈ G and x ∈ X, we sometimes write g(x) for the image of x under the transformation corresponding to g. The set of all g ∈ G such that g(x) = x is called the stabilizer of x. The set {g(x) | g ∈ G} is called the orbit of x. Problems and theorems. 5 5.1. Stabilizers of different elements in the same orbit are conjugate subgroups. 5.2. Let O be the orbit of x under an action of a group G. Denote by H the stabilizer of x. If G is finite, then both H and O are finite. Moreover, |G| = |H| · |O|. 5.3. Let G be a finite group. Suppose that the index of each proper subgroup of G is divisible by p. Then the center of G is non-trivial (i.e. has more than one element). Hint: consider the action of G on itself by conjugations. 5.4. Let p > 0 be a prime. A finite group G is said to be a p-group if |G| = pn for some positive integer n. If G is a p-group, then its center is non-trivial. Every p-group is solvable. 5.5. A subgroup H ⊆ G is called a Sylow’s p-subgroup if |H| is the maximal power of p that divides |G|. Let G be a finite group and p a prime that divides |G|. Then G contains a Sylow’s p-subgroup. Hint: use problem 3 and induction on |G|. 5.6. Sylow’s theorem. Let G be a finite group. Prove the following • Any p-subgroup of G lies in a Sylow’s p-subgroup. • All the Sylow’s p-subgroups are conjugate. • The number of Sylow’s p-subgroups is congruent to 1 modulo p and divides the order of G. Hint: Let S be the set of all Sylow’s p-subgroups. Let G act on S by conjugations. 5.7. Let N be a normal subgroup of G and H another subgroup such that N H = G. Then H acts on N by conjugations. If this action is trivial, and H ∩ N = {1}, then G is isomorphic to N × H. 5.8. Let G and H be two groups. Consider an action ϕ of H on G by automorphisms. Introduce a group structure on G × H by the following rule: (g1 , h1 )(g2 , h2 ) = (g1 ϕ(h1 )(g2 ), h1 h2 ). Show that this is indeed a group structure. The group thus defined is called a semi-direct product of G and H. 5.9. Let N be a normal subgroup of G and H another subgroup such that G = N H and N ∩ H = {1}. Show that G is isomorphism to a semi-direct product of N and H. 5.10. The Classification theorem for finitely generated abelian groups. Any finitely generated abelian group is a direct sum of cyclic groups. Moreover, one can arrange that any summand is isomorphic either to Z or to Z/pk Z for a prime p and a positive integer k. 6