Chapter 7: Groups

Prove that the additive group $\mathbb{Z}\left[x\right]$is isomorphic to the multiplicative group ${\mathbb{Q}}^{\ast \ast }$of positive rationals. [Hint: Let ${\mathit{p}}_{\mathbf{0}}\mathbf{,}{\mathit{p}}_{\mathbf{1}}\mathbf{,}{\mathit{p}}_{\mathbf{2}}\mathbf{,}\mathbf{.}\mathbf{..}$be the distinct positive primes in their usual order. Define$\phi :\mathbb{Z}\left[x\right]\to {\mathbb{Q}}^{\ast \ast }$ by$\phi (a_0+a_{1}x+a_2{x}^2+\cdots +a_n{x}^n)=p_0^{a_0}{p}_1^{a_1}\cdots p_n^{a_n}.]$

Prove that G is an abelian group if and only if$\mathbf{\text{Inn}}\mathit{G}$ consists of a single element.

(a) Verify that the group Inn${\mathbf{D}}_{\mathbf{4}}$ has order 4.

Question: (b) Prove that $\text{Inn}{D}_4\cong {\mathbb{Z}}_2\times {\mathbb{Z}}_2$

Let$f:G\to G$ be given by, $f\left(a\right)=a^{-1}$. Prove that $f$is a bijection.

Find the inverse of the given group element.

(a) $\mathbf{\left(}\begin{array}{cc}\mathbf{2}& \mathbf{0}\\ \mathbf{2}& \mathbf{1}\end{array}\mathbf{\right)}\mathbf{\text{\hspace{0.17em}in\hspace{0.17em}}}{\mathit{\mathbb{Z}}}_{\mathbf{3}}$

(b) $\mathbf{\left(}\begin{array}{cc}\mathbf{1}& \mathbf{2}\\ \mathbf{3}& \mathbf{4}\end{array}\mathbf{\right)}\mathbf{\text{\hspace{0.17em}in\hspace{0.17em}}}{\mathit{\mathbb{Z}}}_{\mathbf{5}}$

(c)$\left(\begin{array}{cc}\mathbf{3}& \mathbf{5}\\ \mathbf{4}& \mathbf{6}\end{array}\right)\mathbf{\text{\hspace{0.17em}in\hspace{0.17em}}}{\mathit{\mathbb{Z}}}_{\mathbf{7}}$

In Exercises 4-8, list (if possible) or describe the elements of the given cyclic subgroup.

5. in the additive group $\mathrm{\mathbb{Z}}$.

Prove that the function$g:{\square }_9\to {\square }_9$ defined by$g\left(x\right)=2x$ is an isomorphism.

Find the order of each permutation.

(a) (12)

Find the order of each permutation.

(b) ( 123 )