# 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 ${\displaystyle e}$ denote the identity element of some group.

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

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

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