Question

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

42)${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$

Short answer

Answer

${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$are not isomorphic.

Step-by-step solution

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 andℤ2×ℤ2×ℤ2are not isomorphic

If ${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$are isomorphic

Then,$\varphi :{\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$

Since the order of ${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$is 4 and the order of ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$is 2.

Hence, there is no element in ${\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$which can generate an element of${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_2$ order 4 in .

Therefore, ${\mathrm{\mathbb{Z}}}_4\times {\mathrm{\mathbb{Z}}}_{2}and{\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2\times {\mathrm{\mathbb{Z}}}_2$are not isomorphic.

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