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.