 # Exercise 1. (Semi-direct product) 1. Let (G,* be a group (this not...

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Exercise 1. (Semi-direct product) 1. Let (G,* be a group (this notation will be valid for the rest of the exercise). Let A < G and B < G, recall what is the product law on A X B. 2. Assume now that A a G and B < G. We denote A X B == {a * b, a € A, b € B}. Show that (A X B) What is explicitely the group law on A X B? The group A X B is called the internal semi-direct product of A and B. 3. Let AG/A the canonical projection. Show that the two following propositions are equivalent : (a) There exists B (b) There exists a section S of TT, i.e. S € Hom (G/A, G) such that TO S - idG/A. 4. Let n € N \ {0,1}. Let t € Sn a transposition. Show that Sn = An X (t). . \ 1 : 5. Let denote Diln,1(K):= : : X € K* , : 1 0 1 Show that GLn(K) = SLn (K) X Diln,1 (K) 6. Let now (A,*A) and (B,*B) be two groups and let T € Hom (B, Aut(A)). We define the following application (A X B) X (A X B) A x B ((a, b), (a', b')) (a Show that is a group law on AxB. This group is called the exteral semi-direct product of B by A relatively to T and is denoted A XT B. 7. With the same notations as the previous question, show that A X {eB} a A XT B. 8. Let A Exercise 2. In this problem K. will always be either R or C and K* either R \ {0} or C \ {0}. Let n € N \ {0} 1. Lemmas on transvections : Definition : Let H be an hyperplane (subspace of dimension n - 1) of K", let f € L(K", K) \ {0} a linear form such that H = {x = 0} and let a € H \ {0}. We call transvection of hyperplane H and line K.a the following linear map T(f,a): Kn x Kn x + f(x)a (a) With the same notations as the definition, show that T(f,a) H = idH and K.a = Im(T(f,a) - Id). (b) With the same notations as the definition and b € K, show that (T(f,a)) - 1 = T(f, a) and that T (f,a)oT(f,b) =r(f,a+b). = 1 0 0 0 1 : (c) Show that there exists a basis of Kn such that the matrix of T(f,a) in this basis is : 0 0 : 1 1 0 1 (d) Show that Vf € L(Kn,K) \ {0}, Va € H \ {0}, T (f,a) € SLn(K). (e) Show that Vu € GLn(K), Vf € L(K", K) \ {0}, Va € H = u(a)) and SO that UOT(f,a) is a transvection of hyperplane u (H) (with H = ker(f)) and line K. u(a). 2. Center of GLn(K) and SLn(K) Definition : We call homothety of Kn coefficient \ the map Xidkn. (a) Let u € GLn(K) such that Va € K", u(K.a) = K.a. Show that u is an homothety. (Hint : split cases between 2 colinear vectors and 2 not colinear vectors). (b) Let u € GLn (IK) commuting with every element of SLn(K). Show that u is an homothety. (c) Show that Z(GLn (K)) = K*. (d) Let us denote un (Ik) == {A € K/X" = 1}. Show that Z(SLn(K)) = SLn(K) n Z(GLn(K)) = un (K). 3. Generators of GLn (IK) and SLn(K) (a) Let x, y € Kn \ {0}. Show that there either exist a transvection u or two transvections u and U such that either u(x) = y or uou(x) = y. (Hint : split cases between 2 colinear vectors and 2 not colinear vectors). i. First case :x and y not colinear. We want to find a transvection u such that u(x) = y, ie u(x) = x + 1(y - x). Let us denote a : = y - x. A. Show that you can find (e3, , en) such that (x, a, e3, , en) is a basis of Kn and let us denote H = Vect (a,e3, , en) Kn K. B. Let us define f Show that f is a linear form and that U = U1X + U2a + Er-3 Uili V1 H = ker (f). C. Consider the transvection T(f,a). Prove it is the map u we were looking for. ii. Second case : A. Show that there exists Z € Kn not colinear to x and y. B. Conclude in this case using the first case. (b) Let H1 and H2 two distinct hyperplanes of Kn and x € Kn \ {0} such that x & H1 u H2. Then there exists a transvection u such that u(x) = x and u(H1) = H2. i. Show that H1 + H2 = Kn ii. Show that dim(H1nh2) = n - 2. Let (e1, en-2, h1) a basis of H1 with h1 € H1 but h1 & H1 nH2. Let (e1, en-2,h2) a basis of H2 with h2 € H2 but h1 & H1 n H2. iii. Let us denote H3 = (H1 nH2) + K.x. Show that H3 is an hyperplane of Kn and that one of its basis is (e1, , en-2,xx. iv. Show that (e1, , en-2, h1,x) is a basis of Kn. { Kn Unx Un -1h1 K v. us now Let define f : U = + - + En-zuei Show that f is a linear form Un-1 and that ker(f) = H3 and f(h1) = 1. Kn vi. Let us denote a := h2 - h1. Let us define u : Show that u is a transvection v= f(v)a of hyperplane H3 and that by definition u(x) = x and = (c) Show by induction on n that the set of all transvections of Kn generate SLn (K). (Hint : for any f € SLn(K), show by last that two questions that up to multiplying by some transvections one can get to the case where f(H) = H and f(x) = x for some hyperplane H of Kn and some x € K" \ H) (d) Show that the set of all transvections of Kn and of all dilatations of Kn generate GLn (K).

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