# 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.