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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