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Any homomorphism from $S_n$ to an abelian group $G$ is given by $\;f(\sigma) = e$, if $\sigma$ is an even permutation, and $f(\sigma)= a$, where order of $a =2$, if $\sigma$ is an odd permutation.

I can prove that this is a homomorphism, but what guarantees that there is no other homomorphism other than this, and why are all even permutations mapped onto the identity of $G$?

I only know that under a homomorphism, identity of $G_1$ goes to identity of $G_2$ and kernels are normal subgroups ($A_n$ are normal), but why there are no possibilities in which kernel is not $A_n$?? I am confused.

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  • $\begingroup$ You seem to be confusing homeomorphism (a topological notion) with homomorphism (an algebraic notion). $\endgroup$ – Bernard Apr 16 '18 at 18:36
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Hint If $f : S_n \to G$ is a group homomorphism and $f((i,j))=a$ than $a^2=e$ in $G$.

Therefore, $f$ takes each transposition in some element of order 2.

Next, if $G$ is abelian, use the fact that $$(i,j)(1,i)(i,j)=(1,j)$$ And $$(1,j)(1,i)(1,j)=(i,j)$$ to deduce that $$f((1,i))=f((i,j))=f((i,j)) \forall (i,j)$$

Finally, write each permutation as a product of transpositions.

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  • $\begingroup$ Iam confused , since then by homomorphism property , O f((i,j) /o((i,j)) => O f(i,j)/2 - O f((i,j)) =1 or 2 , O f ((i,j)) =1 then all 2 cycles are in kernel why this can't happen ?? $\endgroup$ – user534210 Apr 18 '18 at 5:52
  • $\begingroup$ @user534210 If all the 2-cycles are in the Kernel, since any permutation is a product of two cycles, your homomorphism is trivial. $\endgroup$ – N. S. Apr 18 '18 at 15:46
  • $\begingroup$ ohh , nice thank you , but if some of the cycles are in kernel (not all ) then ?? $\endgroup$ – user534210 Apr 18 '18 at 19:59
  • $\begingroup$ @user534210 Let $f(1,2)=a$. Then, by the above you have $a^2=e$ and $f(i,j)=a$ for all $i,j$. If $a=e$ it follows that $f=Id$. Otherwise, writing every permutation as a product of transpositions, and using that $f$ takes each trasposition to $a$, you get that $f(\sigma)=e,a$ is $f\sigma$ is even/odd. $\endgroup$ – N. S. Apr 18 '18 at 20:36
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The group homomorphism provides an isomorphism $S_n/\text{ker}(f)\cong \text{im}(f)$ whereby $\text{ker}(f)$ is a normal subgroup. In this case you have $\ker(f)\in\{1,A_n,S_n\}$. G abelian implies $\ker(f)\neq 1$ and therefore $S_n/\text{ker}(f)\cong 1$ or $S_n/\text{ker}(f)\cong\mathbb{Z}/ 2\mathbb{Z}$

For n=4 there is the addition normal subgroup $V_4$.

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  • $\begingroup$ can you explain me - S-n to G (abelian ) why kernel cannot be identity ??(S-n/{e} is isomorphic to some subgroup of abelian group , => s_n/{e} is abelian group, how we arrive at a contradiction ? plz explain iam a starter in a group theory. $\endgroup$ – user534210 Apr 18 '18 at 5:44
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If such a homomorphism $S_n \to G$ exists, then it induces a surjective homomorphism $S_n \to \langle a \rangle \cong C_2$. Therefore, the kernel has index $2$ and so is $A_n$ because it contains $A_n$.

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