Question

In Exercises 1-9, verify that the given function is a homomorphism and find its kernel.

$\mathit{f}\mathbf{:}{\mathbf{S}}_{\mathbf{n}}\mathbf{\to }{\mathbf{\mathbb{Z}}}_{\mathbf{2}}$, whererole="math" localid="1654585699138" $\mathit{f}\mathbf{\left(}\mathbf{\sigma }\mathbf{\right)}\mathbf{=}\mathbf{0}$if$\mathit{\sigma }$is even and$\mathit{f}\mathbf{\left(}\mathbf{\sigma }\mathbf{\right)}\mathbf{=}\mathbf{1}$ if$\mathit{\sigma }$is odd.

Short answer

It is proved that,$f$is a homomorphism and Ker$f=\{\sigma \in S_n:\sigma iseven\}=A_n$ .

Step-by-step solution

To show f is a homomorphism

Definition of Group Homomorphism

Let $(G,\ast )$and $(G\text{'},\ast \text{'})$be any two groups. A function $f:G\to G\text{'}$is said to be a group homomorphism if $f(a\ast b)=f\left(a\right)\ast \text{'}f\left(b\right),\forall a,b\in G$.

Definition of Kernel of a Function

Let $f:G\to H$ be a homomorphism of groups. Then the kernel of $f$is defined by the set $\{a\in G:f\left(a\right)=e_H\}$, where $e_H$is an identity element.

It is the mapping from elements in role="math" localid="1654586114033" $G$onto an identity element in role="math" localid="1654586117930" $H$by the homomorphism role="math" localid="1654586122429" $f$.

Letrole="math" localid="1654586149158" $f:{\mathrm{S}}_{\mathrm{n}}\to {\mathrm{\mathbb{Z}}}_2$be the function defined as,$\mathrm{f}\left(\mathrm{\sigma }\right)=0$if$\sigma$is even and$f\left(\sigma \right)=1$

ifrole="math" localid="1654586311869" $\sigma$is odd.

We have to show that$f$is a homomorphism.

Let $\sigma ,\tau \in S_n$.

We show that $f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$as follows:

Case1: $\sigma$is even and role="math" localid="1654586410380" $\mathit{\tau }$is even

Since$\sigma$is even androle="math" localid="1654586482826" $\tau$is even, so${\tau }^{-1}$is and therefore $\sigma {\tau }^{-1}$is even.

Then, prove $f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$as follows:

$\begin{array}{c}f\left(\sigma {\tau }^{-1}\right)=0\mathrm{mod}2\\ =0+0\\ =f\left(\sigma \right)+f{\left(\tau \right)}^{-1}\end{array}$

$\therefore f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$.

Case2: role="math" localid="1654586895096" $\mathit{\sigma }$ is even and $\mathit{\tau }$is odd

Since$\sigma$is even and$\tau$is odd so is${\tau }^{-1}$and therefore $\sigma {\tau }^{-1}$is odd.

Then, prove $f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$as follows:

$\begin{array}{c}f\left(\sigma {\tau }^{-1}\right)=1\mathrm{mod}2\\ =0+1\\ =f\left(\sigma \right)+f{\left(\tau \right)}^{-1}\end{array}$

$\therefore f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$

Case3: role="math" localid="1654586952935" $\mathit{\sigma }$is odd and role="math" localid="1654586946019" $\mathit{\tau }$is even

Since$\sigma$is odd and$\tau$is even so is${\tau }^{-1}$and therefore$\sigma {\tau }^{-1}$ is odd.

Then, prove $f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$as follows:

$\begin{array}{c}f\left(\sigma {\tau }^{-1}\right)=1\mathrm{mod}2\\ =1+0\\ =f\left(\sigma \right)+f{\left(\tau \right)}^{-1}\end{array}$

$\therefore f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$

Case 4: $\mathit{\sigma }$is odd and $\mathit{\tau }$is odd

Since$\sigma$is odd and$\tau$is odd so is${\tau }^{-1}$and therefore $\sigma {\tau }^{-1}$is odd.

Then, prove $f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$as follows:

$\begin{array}{c}f\left(\sigma {\tau }^{-1}\right)=1\mathrm{mod}2\\ =1+1\\ =f\left(\sigma \right)+f{\left(\tau \right)}^{-1}\end{array}$

$\therefore f\left(\sigma {\tau }^{-1}\right)=f\left(\sigma \right)f{\left(\tau \right)}^{-1}$

Hence, $f$is a homomorphism.

To find kernel of f

We find kernel of $f$as:

$\begin{array}{c}\text{Ker}f=\{\sigma \in S_n:\sigma iseven\}\\ =A_n\end{array}$

Hence, is a homomorphism and Ker$f=\{\sigma \in S_n:\sigma iseven\}=A_n$ .