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

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