Introduction to Abstract Algebra/Problem set 3

Derivations of Properties
In the following exercises, you are prompted to give proofs which support the statements. Let $$e$$ denote the identity element of some group.


 * 1) Prove that if $$ca = cb$$ or $$ac = bc$$, then $$a = b$$.
 * 2) Prove that if $$abc = e$$, then $$abc = bca = cab = c^{-1}b^{-1}a^{-1} = b^{-1}a^{-1}c^{-1} = a^{-1}c^{-1}b^{-1} = (abc)^{-1}$$.

In the following exercises, you are prompted for proofs supporting the statements regarding various subsets of the real numbers, $$\reals$$. For reference, $$\reals^{+} = \left\{x \in \reals | x > 0\right\}$$ and $$\reals^{-} = \left\{x \in \reals | x < 0\right\}$$.


 * 1) Prove that $$(\reals^{-}, \diamond)$$ is a group where $$a \diamond b = -ab$$ for $$a, b \in \reals^{-}$$.
 * 2) Prove that $$(\reals^{+}, +)$$ does not form a group.
 * 3) Prove that a homomorphism from $$(\reals^{+}, \cdot)$$ to $$(\reals^{-}, \diamond)$$ exists.
 * 4) Prove that a homomorphism from $$(\reals^{-}, \diamond)$$ to $$(\reals^{+}, \cdot)$$ exists.
 * 5) Prove that there is a bijection, $$h: \reals^{+} \rightarrow \reals^{-}$$ for which it is true that $$h(a \cdot b) = h(a) \diamond h(a)$$ and $$h^{-1}(c \diamond d) = h^{-1}(c) \cdot h^{-1}(d)$$. We say that $$h$$ is an isomorphism between the two groups.