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1 1. Let G be a finite, abelian group written additively. Let x = g G g, and let G 2 be the subgroup of G defined by G 2 = {g G 2g = 0}. (a) Show that x = g G 2 g. (b) Show that x = 0 if G 2 = 2. If G 2 = 2, show that x is the unique non-zero element of G 2. (c) Show that G 2 = 2, iff the 2-Sylow subgroup of G is cyclic and non-trivial. (d) Let p be prime. Show that (p 1)! = 1 mod p (Wilson s Theorem). (Hint. Apply part (b) to the multiplicative group (mod p) Z p = {1, 2,..., p 1}.) Solution: (a) x = g G 2 g + g G\G 2 g. Now, in the second summation, every g G \ G 2 has period not equal to 2, and hence g is not an inverse of itself. Moreover, g G \ G 2 = g G \ G 2. Thus, every element in the second sum can be paired of with its inverse, making the sum 0. Hence, x = g G 2 g. (b) Every element other than 0 in G 2 has period 2. Thus, G 2 must be a power of 2 (say 2 m ). Choose m different non-zero elements of G 2, a 1,..., a m. Then, G 2 = a 1 a m (using the theorems of direct sums). Clearly, if m > 1 then summing over all elements of G 2 will produce 0 (just verify component-wise). When m = 1, x = a 1 the unique non-zero element. (c) If G 2 = 2 then the unique 2-Sylow subgroup is non-trivial (has to contain G 2 ) and has exactly one element x of order 2. Now, the 2-Sylow subgroup being an Abelian 2-group is isomorphic to a direct sum Z 2 r 1 Z 2 r k. Let, a 1,..., a k be the generators of the corresponding cyclic groups. But, 2 ri 1 a i has period 2 and must equal x for 1 i k. But, the 2-Sylow subgroup is a direct sum of the a i, and hence a i a j = {0} for i j. Hence, k has to be 1 and the 2-Sylow subgroup must be cyclic. Conversely, if the 2-Sylow subgroup is non-trivial cyclic with generator a of order 2 r, r 1, it has exactly one element of order 2, namely 2 r 1 a, and hence G 2 = 2. (d) The multiplicative group of the field F p is Abelian. Also, the equation x 2 1 has exactly two solutions 1, 1. Thus, applying part (b) we have that the product of all elements is equal to the unique non-identity in G 2 = {1, 1}, which is 1. (Note that, we are using multiplicative notation here and so sums are written as products, the identity is 1 etc., but since the underlying group is Abelian we can use the results of part (a) and (b). 2. G be a finite group, and ϕ : G G a homomorphism. (a) Prove that there exists a positive integer n such that for all integers m n, we have Im ϕ m = Im ϕ n and Ker ϕ m = Ker ϕ n. 1

2 (b) For n as above, prove that G is the semidirect product of the subgroups Ker ϕ n and Im ϕ n. Solution: (a) We have a sequence of subgroups, Ker ϕ Ker ϕ 2 Ker ϕ 3, which must stabilize since G is finite. Similarly, the sequence Im ϕ Im ϕ 2 Im ϕ 3 must stabilize as well. (b) Let x Ker ϕ n Im ϕ n. Then x = ϕ n (y) for some y G, and ϕ n (x) = ϕ 2n (y) = e. But then y Ker ϕ 2n = Ker ϕ n, and so x = e. This shows that Ker ϕ n Im ϕ n = {e}. Since G/ Ker ϕ n Im ϕ n, we have G = Ker ϕ n Im ϕ n. Since (Ker ϕ n )(Im ϕ n ) is a subgroup of G of order G, it must equal G. Consequently G is a semidirect product of Ker ϕ n and Im ϕ n. 3. Let G be a finite group, and p a prime integer dividing G such that the map ϕ : G G, where ϕ(x) = x p, is a homomorphism. (a) Prove that G has a unique p-sylow subgroup P. (b) Prove that there is a normal subgroup N G such that N P = {e} and G = P N. (c) Show that G has a nontrivial center. Solution: (a) By the previous problem, there exists n such that Im ϕ m = Im ϕ n and Ker ϕ m = Ker ϕ n for all m n. But then Ker ϕ n is the subgroup of G consisting of all elements of G whose order is a power of p. Consequently P = Ker ϕ n is the unique p-sylow subgroup of G. (b) Let N = Im ϕ n. By the previous problem, N P = {e} and G = P N. Let x pn N and g G. Then gx pn g 1 = (gxg 1 ) pn N, and so N is normal. (c) Since N G and P G, we have [n, h] N P for all elements n N and h P. But N P = {e}, so [n, h] = e. By the previous problem G is a semi-direct product of P and N but, since elements of N commute with those of P, we actually have G P N. The p-group P has a nontrivial center which is contained in the center of G. 4. Let Q = {±1, ±i, ±j, ±k} be the quaternion group, i.e., ( 1) 2 = 1 is the identity element, i 2 = j 2 = k 2 = 1, and ij = k = ji, jk = i = kj, ki = j = ik. (a) Determine all subgroups of Q and prove that they are normal. (b) What is the order of Aut(Q)? Solution: (a) Aside from e, the group Q consists of six elements of order 4 and one element of order 2, namely 1. Consequently the only subgroups of Q are Q, i, j, k, 1, {e}. 2

3 (b) Any two elements of order four, which are not powers of each other, constitute a generating set for Q. Any automorphism ϕ Aut(Q) is determined by its behavior on a generating set, and must take a generating set to a generating set. Consider the generating set {i, j}. Then ϕ(i) {±i, ±j, ±k} and ϕ(j) {±i, ±j, ±k} \ {±ϕ(i)}. Consequently Aut(Q) = Let p < q be prime numbers such that p divides q 1. If G is a non-abelian group of order pq, show that Z(G) = {e}. Solution: If Z(G) is nontrivial, then Z(G) equals p or q. But then G/Z(G) is cyclic, in which case G must be abelian. 6. Determine, up to isomorphism, all groups of order 63. Solution: Let G = 63. Since 63 = 3 2 7, by an earlier homework problem either the 3-Sylow or the 7-Sylow subgroup is normal in G. Suppose the 3-Sylow subgroup P is normal, then G P α Z/7 for α : Z/7 Aut(P ). If P Z/9 then Aut(P ) = 8 and α is trivial. In this case, G Z/9 Z/7 Z/63. If P Z/3 Z/3 then Aut(P ) = 48 and once again α must be trivial. This implies that G Z/3 Z/3 Z/7 Z/3 Z/21. If P is not normal then the 7-Sylow subgroup Q Z/7 is normal, and so G Z/7 β P where β : P Aut(Z/7) is a nontrivial homomorphism. Note that Aut(Z/7) Z/6, and so it has a unique subgroup of order 3, which must be the image of β. Consequently, up to isomorphism, there are exactly two other groups of order 63, namely G Z/7 β Z/9 and G Z/7 β (Z/3 Z/3). 7. Recall the definitions of fibre product and fibre co-product. (a) Show that fibre products exist in the category of Abelian groups. If f : X Z, g : Y Z are Abelian group homomorphisms, show that X Z Y is the set of all pairs (x, y), x X, y Y such that f(x) = g(y). (b) Prove that the pull back of a surjective homomorphism is surjective. 8. The purpose of this problem is to show that fibre coproducts exist in the category of Abelian groups. 3

4 (a) In this case the fibre coproduct of two homomorphisms, f : Z X, g : Z Y is the factor group, X Z Y = (X Y )/W, where W is the subgroup consisting of all elements (f(z), g(z)), z Z. (b) Prove that the push out of an injective homomorphism is injective. Solution: Omitted. 9. A groups is called finitely generated (resp. finitely presented) if it admits a presentation, (t i ) i I ; (r j ) j J where I is a finite set (resp. where I and J are finite sets). (a) Let G be a finitely presented group and x = (x α ) α A a finite generating family of G. Show that there exists a finite family s of relators of the family x such that x; s is a presentation of G. Solution: Let t; r be a finite presentation of G, where t = (t i ) i I, and r = (r j ) j J and I, J are finite sets. Let T = (T i ) i I (where T i are new symbols) and let r j (T ) denote the words with the new symbols T substituted for t. Similarly, let X = (X α ) α A. For any set S, let F [S] denote the free group on S. Let R t be the normal subgroup of F [T ] generated by r(t ). Then, G = F [T ]/R t. Let φ : F [T ] G be the homomorphism sending T i to t i. Clearly, R t is the kernel of φ. Similarly, ψ : F [X] G, sending X α to x α. Let R x be the kernel of ψ. For each i I, let A i F [X], such that t i = ψ(a i ). Let γ 1 : F [T ] F [X] sending T i to A i. Similarly, for each α A, let B α F [T ] such that x α = φ(b α ) and let γ 2 : F [X] F [T ] sending X α to B α. We claim, that R x is the normal subgroup generated by the elements r j (A), j J and Xα 1 γ 1 (γ 2 (X α )), α A. Let R x be the normal subgroup of F [X] generated by these elements. Then, clearly R x R x. The homomorphism γ 1, after passing to the quotient gives a homomorphism G = F [T ]/R t F [X]/R x, which is inverse to the canonical homomorphism F [X]/R x F [X]/R x = G. This shows that R x = R x. (b) Show that a finite group is finitely presented. Solution: The finite set of elements of the group and the finite multiplication table gives a finite set of generators and relators respectively. (c) Given an example of a finitely presented group with a subgroup which is not finitely generated. Solution: Let S = {a, b} and G = F [S] the free group generated by S. Let (w k ) k 0 be any enumeration of the reduced words of G, which begin and end with 4

5 powers of a. Let B = {b k w k b k k 0}. Let H be the subgroup of G generated by B. We claim that there is no non-trivial relations on the elements of B and hence H is free on B and is thus not finitely generated. Clearly, a word on the elements of B could reduce to identity iff it is of the form s ɛ 1... s ɛn where s = b k w k b k for some fixed k and ɛ i {±1}. However, in this case it is easy to see that it must be the trivial relation. (d) If G is a finitely presented group and H a normal subgroup of G, show the equivalence of the following properties: i. G/H is finitely presented. ii. There exists a finite subset X of G such that H is the normal subgroup of G generated by X. Solution: Suppose that G/H is finitely presented. Let x = (x i ) i I and s = (s j ) j J be the generators and relators of G. Similarly, let u = (u α H) α A and r = (r β ) β B be the generators and relators of G/H. Here, I, J, A, B are all finite sets. Let X = (X i ) i I and U = (U α ) α A. Consider the homomorphisms, F [U] G G/H, where the first homomorphism is defined by U α u α and the second homomorphism is the canonical homomorphism. Clearly, the kernel of the composition is the normal subgroup of F [U] generated by the elements r β (U). Hence, the images of these elements in G must generate H. Hence, H is generated by the finite set of elements r β (u). Conversely, suppose that there exists a finite set of elements g = (g k ) k K which generate H. For each k K, let W k F [X] such that w k = g k. It is easy to see that G/H can be described by the generators (x i H) i I, and relators (s j ) j J and (W k ) k K. 10. Show that the group defined by the presentation x, y; xy 2 (y 3 x) 1, yx 2 (x 3 y) 1 is the trivial group. Solution: We first show that x 2 y 8 x 2 = y 18 and x 3 y 8 x 3 = y 27. xy 2 = y 3 x = xy 2 x 1 = y 3. Squaring and cubing both sides we get, xy 4 x 1 = y 6,...(i) 5

6 and xy 6 x 1 = y 9...(ii) Substituting (i) into (ii) we get, x 2 y 4 x 2 = y 9,...(iii) and squaring, x 2 y 8 x 2 = y 18...(iv) Multiplying (iii) and (iv) we get, Also, squaring (i) we have that, Substituting (vi) into (v) we get that, Now, x 2 y 12 x 2 = y 27...(v) xy 8 x 1 = y 12...(vi) x 3 y 8 x 3 = y 27...(vii) y 27 = x 3 y 8 x 3 = (x 3 y)y 8 (y 1 x 3 ) = yx 2 y 8 x 2 y 1 = yy 18 y 1 = y 18. This implies that y 9 = e. Now, y 2, y 3 are conjugates and hence have the same period. Hence, y = e and hence x = e. 6

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