Question

Question:In Exercise 40-44, Explain why the given groups are not Isomorphic .(Exercises 16 and 29 may be helpful.)

41)${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{D}_4$

Short answer

Answer

${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{D}_4$ are not isomorphic.

Step-by-step solution

 Step 1: Definition of Isomorphism

$$\begin{gathered} \text{LetGandHbegroupswithagroupoperationdenotedby*,} \\ \text{GisisomorphictoagroupH(insymbolsG≅H)} \\ \text{ifthereisafunctionf:→Hsuchthat,} \\ 1.If(a*b)=f\left(a\right)*f\left(b\right) \\ 2.fisinjective. \\ 3.fissurjective. \end{gathered}$$

Showing that ℤ4×ℤ2 and D4 are not isomorphic

If ${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{D}_4$are isomorphic

Then,$\varphi :{\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2\to D_4$

Since${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$isan abelian group,$D_4$is not. So, they are not isomorphic.