Question

Question: Is is isomorphic U₈to U₁₂? Justify your answer.

Short answer

Answer

U₈ and U₁₂are 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 U8 and U12 are isomorphic

If U₈and U₁₂are Isomorphic.

Then,$\varphi :U_8\to U_{12}$

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, order of the elements U₈ in is 2.

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 in both U₈ andU₁₂ are same so, every element in U₁₂ can be produced by U₈ . Hence, they are isomorphic.

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