Question

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

43) U₈and U₁₀

Short answer

Answer

U₈and U₁₀are not Isomorphic.

Step-by-step solution

 Step 1: Definition of Isomorphism

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

Showing that U8  and U10 are not isomorphic

If U₈and U₁₂are Isomorphic.

Then,

U₈ ={1,3,5,7}

The order of the element in U₈ can be given as:

1¹ = 1

3²= 9 = 1

5² = 25 = 1

7² =49 = 1

So, the order of the elements U₈ in is 2.

The order of elements in

U₁₀= {1,3,7,9}

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

From the above, it can be seen that the order of at least one element inU₁₀ is 4.

Therefore, U₈ and U₁₀ are not isomorphic.