# Solutions to Problem Set 5

Math 345/504

2-5-2018

1. If the function is a homomorphism, prove it. If it is not, give a specific counterexample which shows that it is not.

(a) The function given by

( is the nonzero real numbers; is the positive real numbers.)

(b) The function defined by

(The trace of a square matrix A is the sum of the entries on the main diagonal of A. is a group under matrix addition.)

(c) The function given by

( is a group under vector addition.)

(a) Let . Then

Therefore, f is a group map.

(b) Consider the following elements of :

Then

Therefore, , and g is a group map.

(c)

Therefore, , so h is not a group map.

2. is the group of real numbers under addition, and is the group of positive real numbers under multiplication.

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

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

(c) Prove that f and g are isomorphisms by showing that f and g are inverses: and for all x for which the composites are defined.

(a)

(b)

(c)

Courage consists of the power of self-recovery. - Ralph Waldo Emerson

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