MAT301 Assignment 2 Show work in all problems. (1) Let E(n) be the group of all symmetries of Rn ( not necessarily fixing 0). Prove that E(n) = {f : Rn → Rn | f(u) = Au + b where A ∈ O(n), b ∈ Rn}. (2) Let G be a group and let a, b, c ∈ G such that a5 = b4 = c7 = e. Express ((a2b−3)−2c3)−2 as a product of positive powers of a, b, c. (3) Let G be a group such that every non identity element of G has order 2. Prove that G is abelian. (4) Suppose a subset H in a group G has the property that for any a, b ∈ H implies that b−1a−1 ∈ H also. Does this imply that H is a subgroup of G? Hint: Look at G = Zm for small m. (5) Let (G, ·) be a group and. H,K ⊂ G be subgroups. Recall that HK = {h · k | h ∈ H, k ∈ K}. Is HK always a subgroup of G? If yes, prove it. If not, give a counterexample. (6) (5 pts) Let G = (Z,+) and let H = 〈18, 24, 42〉 be the subgroup generated by 18, 24, 42. Find d > 0 such that H = 〈d〉. (7) Let l be a line through 0 in R2 and α > 0 be any angle. Let Fl be the reflection nin l and Rα be the rotation by α around 0. Let H = 〈Rα, Fl〉 ⊂ O(2) be the subgroup of O(2) generated by Rα and Fl. Prove that H = {F kl Rnα | k = 0, 1, n ∈ Z}. Hint: Show that Fl ◦Rα = R−α ◦ Fl. (8) Let G = SL(2,R) and let A = ( 0 1 −1 2 ) . Find the centralizer of A in G. (9) Let n ≥ 2 and let H ⊂ Dn be a subgroup. Prove that either every element of H is a rotation or exactly half of the elements of H are rotations. 1 2Hint: Use that the product of two rotations is a rotations, the product of two reflections is a rotation and the product of a rotation and a reflection is another reflection.