# Solutions to Problem Set 15

Math 345/504

3-7-2018

1. (a) Find the order of in .

(b) List the elements of the cyclic subgroup of generated by .

(c) Compute the product in the group . (Multiply permutations right to left, and write the result in disjoint cycle form.)

(a) The order of 20 in is . The order of 44 in is . Therefore, the order of in is .

(b)

(c)

2. List the elements of the cyclic subgroup in .

3. Find a specific element of order 10 in .

Note that . I'll find an element of order 2 in and an element of order 5 in .

There one element of order 2 in , namely 4.

There are several elements of order 5 in ; probably the simplest is 7.

Hence, is an element of order 10 in .

4. is a group under addition. Define by

(a) Prove that f is a group map.

(b) Prove that

(a)

(b)

Therefore, .

Suppose . Then

Therefore, .

This proves that .

[Math 504]

5. Find an element of having the largest possible order. Explain why no element can have larger order.

If , then . So every element of has order less than or equal to 30.

The element has order 30 in . Therefore, it's an element of the largest possible order.

6. Consider the following subset of :

Either prove that H is a subgroup of , or give a specific counterexample to one of the subgroup axioms.

H is not a subgroup.

H is not closed: , but

Alternatively, , but , contradicting closure under inverses.

The way of progress is neither swift nor easy. - Marie Curie

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