Question

Question: The function $f:U_5\to U_5$ given by is a homomorphism by $f\left(x\right)=x^2$exercise 23. Find k_f( notation as in exercise 33)

Short answer

Answer

k_f= { 1,4}

Step-by-step solution

 Step 1: Definition of the kernel Kf

$$\begin{gathered} \mathbf{\text{Let}}\mathbf{f}\mathbf{:}\mathbf{G}\mathbf{\to }\mathbf{H}\mathbf{}\mathbf{\text{bea}}\mathbf{homomorphism}\mathbf{}\mathbf{of}\mathbf{}{\mathbf{\text{groupthen,thekernalk}}}_{\mathbf{f}}\mathbf{\text{offisasetwrittenas}} \\ {\mathbf{\text{ifK}}}_{\mathbf{f}}{\mathbf{\text{={g∈G|f(g)=e}}}_{\mathbf{H}} \end{gathered}$$

 Step 2: Finding  kf notation

Since f (x) = x²

U₅={1,2,3,4,}

For $f:U_5\to U_5$, we can write the order of elements of as:

$$\begin{gathered} 1·1=1=1 \\ 2·2=·4=4 \\ 3·3=9=4 \\ 4·4=16=1 \end{gathered}$$

As the orders of elements of U₅ are either 1 or 4, there are only two elements in $f:U_5\to U_5$, which are 1 and 4. Hencwe arrive at .