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 :
Therefore, , and g is a group map.
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.
Courage consists of the power of self-recovery. - Ralph Waldo Emerson
Bruce Ikenaga's Home Page
Copyright 2018 by Bruce Ikenaga