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.
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
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.
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 .
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