- 2 13 Marks A Z 1 2 4 7 8 11 13 14 Is A Group Under Multiplication Modulo 15 I Find All Of Its Subgroups 1 (83.01 KiB) Viewed 27 times
2. [13 Marks] a) Z₁ = {1,2,4,7, 8, 11, 13, 14} is a group under multiplication modulo 15. (i) Find all of its subgroups. (ii) Find all the distinct cosets of the subgroup generated by 11. b) Z (the integers) is a subgroup of Q (the rationals) under addition, and Q is a subgroup of R (the reals) under addition. (i) Show that Q/Z has an element of order 2, and that R/Q does not. (ii) Using this, or by some other method, prove that Q/Z and R/Q are not isomorphic. c) Recall that the center of a group G is {g in G: gr = æg for all r in G}; that is, it's the set of elements of G that commute with every element of G. (i) Suppose the element g of G, g 1, is contained in every nontrivial subgroup of G. Prove that the center of G is not trivial. (ii) Suppose N is a normal subgroup of a group G, and the intersection of N with the commutator subgroup of G contains just the identity element. Prove that N is contained in the center of G.