Math 345/504

2-7-2018

1. The function defined by is a group map. List the elements of and the elements of .

2. Find the quotient and remainder when the Division Algorithm is applied to:

(a) Divide 937 by 28.

(b) Divide -937 by 28.

(a)

(b)

3. An integer n is * even* if . Using * only this definition and
properties of divisibility*, prove that if n is even, then is even.

Suppose n is even, so for some . Then

Therefore, is even.

4. Find the largest power of 7 that divides . * Explain your reasoning.*

There are 7 numbers in the set that are divisible by 7 by not :

There is one number in the set that is divisible by , namely 49.

Hence, the largest power of 7 that divides is .

5. Let denote the group of real numbers with the operation of addition. Let denote the group of real numbers under the operation

(a) Prove that given by is a group map.

(b) Show that f is an isomorphism by constructing an inverse for f.

(a) If , then

Therefore, , and f is a group map.

(b) Define by

Then

Thus, f and g are inverses, and f is an isomorphism.

Note: This problem gives an example of two isomorphic group structures on the same set.

* [MATH 504]*

6. is a group map that satisfies . Find a formula for .

Note that in . So

It follows that .

7. Prove, or disprove by specific counterexample: "If and and , then ."

The statement is false, since and , but .

*You probably wouldn't worry about what people think of you if you
could know how seldom they do.* - *Olin Miller*

Copyright 2018 by Bruce Ikenaga