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Page 113 - Dummit and Foote - Group actions

Two group elements induce the same permutation on $A$ if and only if they are in the same coset of the kernel.

What does this mean? Two permutations induce the same permutation on $A$ looks like it means $\sigma_1 A = A' = \sigma_2 A$ and then I don't really understand the right hand side of the $\iff$.

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  • $\begingroup$ Could you provide more context? What group action are you dealing with and which kernel is meant here? $\endgroup$ – Stefan Hamcke Aug 12 '15 at 7:35
  • $\begingroup$ @StefanHamcke I don't know what is relevant context unfortunately, it just came up in D&F and I had read all the previous parts of the chapter, that seem unrelated $\endgroup$ – Galois in the Field Aug 12 '15 at 9:11
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This is my understanding of the text:

Let group $G$ act on $A$ as $g.a$ $\forall g \in G$ and $a\in A$

Let $\phi: G \to S_A$ be the associated permutation representation (homomorphism)

Let $K$ be the kernel of this map, which is also the kernel of the group action.

Then according to me, the text says

$\phi(g)= \phi(h) \iff gK = hK$

$$\phi(g)=\phi(h)$$

$$ \iff \phi(g)(a)=\phi(h)(a)\; \forall a\in A$$

$$\iff g.a=h.a \; \;\forall a \in A$$

$$\iff h^{-1}.(g.a)= (h^{-1}).(h.a) \; \; \forall a \in A$$

$$\iff (h^{-1}g).a=(h^{-1}h).a=1.a=a \; \;\forall a \in A$$

$$\iff h^{-1}g \in K$$

$$\iff gK = hK$$

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Okay so $A$ is a set being acted upon by a group $G$ (as a group of permutations); the kernel of the action of $G$ on $A$ is defined to be the subgroup fixing $A$, that is the elements $\phi$ such that $a^{\phi} = a$ for all $a \in A$.

Now, suppose that $\sigma_{1}$, $\sigma_{2} \in G$ with $a^{\sigma_{1}} = a^{\sigma_{2}}$ for all $a \in A$ (this is what it means for them to induce the same permutation on $A$)? What can we say about $a^{\sigma_{1} \sigma_{2}^{-1}}$?

On the other hand, suppose we have $\phi$ in the kernel of the action, so $a^{\phi} = a$ for all $a \in A$; what can we say about $a^{\sigma}$ and $a^{\phi \sigma} = (a^{\phi})^{\sigma}$?

(I prefer this exponent notation for group actions, because it makes it clear that $a$ is not considered as an element of $G$, rather it is some object that the elements of $G$ act on.)

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