Does there exist a homomorphism?

Groups, Rings, Domains, Modules, etc, Galois theory
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Tolaso J Kos
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Does there exist a homomorphism?

#1

Post by Tolaso J Kos »

Examine if there exists a homomorphism $\phi:\mathcal{S}_3 \rightarrow \mathcal{S}_5$ that is neither the trivial nor the identity where $\mathcal{S}_3 , \;\mathcal{S}_5$ are the $3$-permutation group and $5$- permutation group, respectively.

Do the same question for $\phi:\mathcal{S}_5 \rightarrow \mathcal{S}_3$.
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Demetres
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Re: Does there exist a homomorphism?

#2

Post by Demetres »

Of course there is. For example take $\varphi(\pi) = (1\,2)(3)(4)(5)$ if $\pi$ is an odd permutation and $\varphi(\pi) = (1)(2)(3)(4)(5)$ if $\pi$ is an even permutation.

Similar example works for the other question.
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