Math 3033 Assignment 4 1. Let G = (Z/18)∗ . (a) List all orders of all elements of G. (b) Is G cyclic? If so list all possible generators. (c) Find all the subgroups of G. 1 2 3 4 2. Write the permutation 2 4 1 5 5 3 as a product of transpositions. 3. Define a function φ by setting φ(n) to be the number of generators of the cyclic group Z/n. (a) If p is prime, then what is φ(p)? (b) For every divisor d of 50, how many elements are there of order d in Z/50. (c) Prove the following formula n= X φ(d). d|n Here are some helpful observations. i. Every element generates a cyclic subgroup. ii. There is exactly one cyclic subgroup of order d for each divisor of n. iii. The cyclic subgroup of order d has φ(d) generators. 4. Draw the Hasse diagram of all the subgroups of Z/100. 5. Is (Z/15)∗ cyclic? Draw the Hasse diagram of all the cyclic subgroups of (Z/15)∗ . 6. (a) Show that if H and K are subgroups of G then H ∩ K is a subgroup of G. (b) Let G = Z and let H = h24i and let K = h18i. Find a generator for the subgroup H ∩ K. (c) If hmi, hni are subgroups of Z find a formula for a generator of hmi ∩ hni.

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