# Tagged Questions

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### Relationship between group actions and homomorphisms

I know that there exist no nontrivial homomorphism from $S_3$ into $Z_5$ as they are groups of co-prime order. I am not looking for an explanation of this but for an explanation concerning the obvious ...
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### Defining a map based on a group action on left cosets

If $H$ is subgroup of $G$ such that the index of $H$ in $G$ is $n$ and $\pi_H$ is the permutation representation of the action of $G$ on the left cosets of $H$, is $\pi_H$ a map from $H$ to $S_n$? I ...
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### Group Action - Permutation on the Polynomial

I'm trying to check the permutation on the polynomial is a Group Action, but I'm not getting the second axiom. I'm following my lecturer's work --- Examples 2.1 and 2.6 on page 5 on ...
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### A step in the proof of Cauchy's theorem for groups

Let $G$ a group, $p$ a prime and $X=G^p$. Let $\sigma\in S_X$ act as follows: $\sigma(x_1,...,x_p) = (x_2,...,x_p,x_1)$. Let $Y$ the subset of elements in $X$ such that $x_1x_2...x_p=1$ and let ...
### Injective Homomorphism on $S_n$
I am working on a project in my group theory class to find an outer automorphism of $S_6$, which has already been addressed at length on this site and others. I have a prescription for how to go about ...