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Mathematics 366 Problems A. Hulpke 1) Show that a factor group of a cyclic group is cyclic. 2) Let G be a group and A, B C G. Show that A ∩ B C G. 3) Let F = Z5 and Q = F[x]/(x2 + 3). Let z be the image of x in Q under the natural homomorphism and calculate z + 1/z. 4) Let G be the symmetry group of a tetrahedron. Determine the stabilizer of a face and representatives of the cosets. 5) Let G = hai be a finite cyclic group and H ≤ G. Let m > 0 be minimal, such that am ∈ H and let b = ak ∈ H. Assume that k = qm + r. Show that ar ∈ H. Conclude that H = ham i. (This shows that every subgroup of a cyclic group is cyclic.) 6) Let G be a group. We define the center of G as Z := {x ∈ G | xg = gx for all g ∈ G} (Note that the definition differs from that of the centralizer: g is not a fixed element, but the condition must hold for every g ∈ G.) a) Show that Z ≤ G. b) Show that Z C G. 7) Let R = {2 · a | 0 ≤ a ≤ 14} = {0, 2, 4, 6, 8, . . . , 28} with arithmetic modulo 30. a) Show that 16 is the identity element of R. (Hint: You must show 2 · a · 16 mod 30 = 2 · a.) b) Determine the inverse of 8 in R. c) Let U be the group of units in R. Is U cyclic? 8) Let G be a group and N C G with [G : N] = 2. Also let H ≤ G with N 6≤ H. Show that [H : H ∩ N] = 2. 9) Let a = (1, 3, 5, 2) and b = (1, 5)(2, 4). Calculate b−1 ab and a−1 ba. 10) Let G be a group, H ≤ G and x ∈ G. We define a set H x by: H x := xHx−1 = {xhx−1 | h ∈ H} Show that H x ≤ G. 11) Let N C G and A C G/N. Let M = {g ∈ G | Ng ∈ A}. Show that M C G and that N ≤ M. 12) Let G = h(1, 2, 3, 4, 5), (2, 3)(4, 5)i be the group of symmetries (rotations and reflections) of a pentagon and H = h(2, 3)(4, 5)i. Determine representatives for the cosets of H in G.