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Mathematics 676-3 Homework (due Mar 3) A. Hulpke 18) Show that an iterated subdirect product of simple groups must be a direct product. 19) Let G = TransitiveGroup(24, 40) = h(1, 11, 9, 19, 17, 3)(2, 12, 10, 20, 18, 4)(5, 15, 14, 24, 22, 7)(6, 16, 13, 23, 21, 8), (1, 14)(2, 13)(3, 4)(5, 17)(6, 18)(9, 22)(10, 21)(11, 12)(19, 20)i . Find all subgroups StabG (1) ≤ S ≤ G. (Hint: Use AllBlocks in GAP.) 20) a) Let n = a + b with a < b. Show that Sn acts primitively on the a-element subsets of n. b) Show that Sa × Sb is a maximal subgroup of Sn . c) Why is a) and b) false if a = b? 21) Let G be transitive on Ω with two unrefinable block systems (i.e. any finer block system is the trivial system B1 ). Show that G can be written as a subdirect product of groups of smaller degree. 22) Let F be a finite field and G = GLn (F), acting on the nonzero vectors of F n . a) Show that this action is imprimitive, by describing a nontrivial block system. b) Describe (e.g. by describing the matrices) the stabilizer of a vector, and the stabilizer of one of a block. c) Show that the action of G on the blocks is primitive. (Thus the block stabilizer is a maximal subgroup of G.) 23) Let Ω = {1, . . . , n}, G ≤ SΩ be a transitive permutation group and S = StabG (1). Let ∆ = {ω ∈ Ω | ωg = ω∀g ∈ S} . a) Show that ∆ is a block for G on Ω. b) Show that StabG (∆) = NG (S).