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 Post subject: True or false statements Posted: Wed Nov 29, 2017 9:20 am

Joined: Sat Nov 14, 2015 6:32 am
Posts: 159
Location: Melbourne, Australia
Let $n \in \mathbb{Z}$ such that $n \geq 2$. Let $\mathcal{S}_n$ be the permutation group on $n$ letters and $\mathcal{A}_n$ be the alternating group. We also denote $\mathbb{C}^*$ the group of non zero complex numbers under multiplication.

Which of the following are correct statements?

1. For every integer $n \geq 2$ there is a non trivial homomorphism $\chi: \mathcal{S}_n \rightarrow \mathbb{C}^*$.
2. For every integer $n \geq 2$ there is a unique non trivial homomorphism $\chi:\mathcal{S}_n \rightarrow \mathbb{C}^*$.
3. For every integer $n \geq 3$ there is a non trivial homomorphism $\chi:\mathcal{A}_n \rightarrow \mathbb{C}^*$.
4. For every integer $n \geq 5$ there is non trivial homomorphism $\chi:\mathcal{A}_n \rightarrow \mathbb{C}^*$.

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$\displaystyle \sum_{n=1}^{\infty}\frac{1}{n^s}= \prod_{p \; \text{prime}}\frac{1}{1-p^{-s}}$

Top   Post subject: Re: True or false statements Posted: Thu Dec 28, 2017 7:09 pm
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Joined: Mon Nov 09, 2015 1:52 pm
Posts: 426
i. True statement.

If $\displaystyle{n\in\mathbb{N}\,,n\geq 2}$, then ,we define $\displaystyle{\mathcal{x}:S_n\to \mathbb{C}^{\star}}$ by

$\displaystyle{\mathcal{x}(\sigma)=1}$ if $\displaystyle{\sigma}$ is an even permutation and

$\displaystyle{\mathcal{x}(\sigma)=-1}$ if $\displaystyle{\sigma}$ is an odd permutation.

The homomorphism $\displaystyle{\mathcal{x}}$ is called sign homomorphism.

iv. False statement.

Suppose that iv. is true. Then, if $\displaystyle{n\in\mathbb{N}\,,n\geq 5}$, we choose a

non trivial homomorphism $\displaystyle{x_{n}:A_{n}\to \mathbb{C}^{\star}}$, that is $\displaystyle{\rm{Ker}(x)\neq A_{n}}$.

Since $\displaystyle{\left(A_{n},\circ\right)}$ is a simple group and $\displaystyle{\rm{Ker}(x)\trianglelefteq A_{n}}$

we get $\displaystyle{\rm{Ker}(x)=\left\{Id\right\}}$, which means that $\displaystyle{A_{n}\cong Im(x)\leq \mathbb{C}^{\star}}$.

Therefore, $\displaystyle{A_{n}}$ is cyclic, a contradiction.

So, there exists $\displaystyle{m\in\mathbb{N}\,,m\geq 5}$ such that, the only homomorphism

$\displaystyle{x:A_{m}\to \mathbb{C}^{\star}}$ is the trivial one.

iii. False statement

We choose the above $\displaystyle{m\in\mathbb{N}\,,m\geq 5>3}$.

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