Question

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

43) U₁₀and U₁₂

Short answer

Answer

U₁₀and U₁₂are 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   U10 and U12 are isomorphic

If U₈and U₁₂are Isomorphic.

$$\begin{gathered} If{U}_{10}and{U}_{12}areIsomorphic.then\varphi :U_{10}\to U_{12} \\ Then,U_{10}=\{1,3,5,7,9\} \end{gathered}$$

So, the order of the elements in U₁₀can be given as

3¹ = 1

3² = 9

3³ = 27 (mod 10) =7

3⁴ = 81(mod 10) =1

The order of elements in

U₁₂= {1,5,7,11}

So, the order of the elements in U₁₂is also 2.

1¹ = 1

5² = 25 = 1

7² =49 = 1

11² =121 = 1

Since order of the elements U₁₂ is 2.and U₁₀is 4. Therefore, no element in U₁₂ can produce elements inU₁₀ .Hence, they are not isomorphic.