# Solutions to Problem Set 10

Math 345/504

2-23-2018

1. (a) Find the order of 92 in .

(b) Find an element of order 35 in .

(a)

(b) in . Since any smaller multiple of 4 will be less than 140, the order of 4 in is 35.

2. Let G denote the following cyclic group of order 155:

Find the order of in G.

3. (a) List the generators of (which is a group under addition mod 15).

(b) List the elements of the cyclic subgroup of (which is a group under addition mod 30).

(c) Using the results of (a) and (b), list the generators of the cyclic subgroup of .

(a) The generators are the elements that are relatively prime to 15:

(b) Note that , so the subgroup is cyclic of order 15.

(c) The generator 1 in corresponds to the generator 4 in , so just multiply the elements in (a) by 4:

4. Consider the following subgroup of :

Since H is cyclic, it must have a generator.

Prove that .

If , then . Therefore, .

Let . Now

Thus, , so .

Hence, .

[MATH 504]

5. Consider the following subgroup of :

Since H is cyclic, it must have a generator.

Prove that .

If , then , , and . Since , the last two divisibility relation implies . Therefore, , and .

Conversely, if , then

Thus, , , and , so . Therefore, , and hence .

There is no greater agony than bearing an untold story inside you. - Maya Angelou

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