Chapter 7: Groups

Is the set ${\mathit{B}}_{\mathbf{n}}$ of odd permutation in${\mathit{S}}_{\mathbf{n}}$ a group? Justify your answer.

10. Show that$\left(1,0\right)$ and$\left(0,2\right)$ generate the additive group $\mathrm{\mathbb{Z}}\times {\mathrm{\mathbb{Z}}}_7$.

Show that the function$f:\square \to \square$ defined by$f\left(x\right)=x^2$ is not a homomorphism.

10. Find the order of every element in each group.

(a)${\mathrm{\mathbb{Z}}}_4$

10. Find the order of every element in each group.

(b)${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$

10. Find the order of every element in each group.

(c)$S_3$

10. Find the order of every element in each group.

(d)$D_4$

10. Find the order of every element in each group.

(e)$\mathrm{\mathbb{Z}}$

Let G be an additive group. Write statement (2) of Theorem 7.8 and statements (1 )-(3) of Theorem 7.9 in additive notation.

Consider the additive group${\mathbb{Z}}_2$ and the multiplicative group $\mathit{L}\mathbf{=}\mathbf{\{}\mathbf{\pm }\mathbf{1}\mathbf{,}\mathbf{\pm }\mathbf{i}\mathbf{\}}$ofcomplex numbers. Write out the operation table for the group${\mathit{\mathbb{Z}}}_{\mathbf{2}}\mathbf{\times }\mathit{L}$ .