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, $ℝ$. For reference, ${ℝ}^{+}=\{x\in ℝ|x>0\}$ and ${ℝ}^{-}=\{x\in ℝ|x<0\}$.

1. Prove that $({ℝ}^{-},\diamond )$ is a group where $a\diamond b=-ab$ for $a,b\in {ℝ}^{-}$.
2. Prove that $({ℝ}^{+},+)$ does not form a group.
3. Prove that a homomorphism from $({ℝ}^{+},\cdot )$ to $({ℝ}^{-},\diamond )$ exists.
4. Prove that a homomorphism from $({ℝ}^{-},\diamond )$ to $({ℝ}^{+},\cdot )$ exists.
5. Prove that there is a bijection, $h:{ℝ}^{+}\to {ℝ}^{-}$ 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.