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I'm refreshing some basic group theory so excuse in advance the basic questions. In $S_3$ one can isolate the alternating group $A_3$ by looking at the even permutations. Is it true in general that the alternating group $A_n$ can be isolated as the group of all even permutations of $S_n$?

In addition is the set of all corresponding permutation matrices in $GL_n(\mathbb{R})$ also a normal subgroup of $GL_n(\mathbb{R})$.

Thanks.

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    $\begingroup$ Isn't that how the alternating group is defined? $\endgroup$ – verret Nov 29 '19 at 23:26
  • $\begingroup$ Ah yes it is the definition! I was reading something else and skipped over the definition! Thanks. Now so now $A_n$ is always the kernel of a homomorphism, right? So it's a normal subgroup, right? $\endgroup$ – Rachid Atmai Nov 29 '19 at 23:28
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Is it true in general that the alternating group $A_n$ can be isolated as the group of all even permutations of $S_n?$

Yes, as it is one way to define the alternating group $A_n$.

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  • $\begingroup$ Thanks Shaun. On a different note also, my understanding is that it's always a normal subgroup because it can be realized as the kernel of a homomorphism, generalizing the $A_3$ and $S_3$ set up. Is that the only normal subgroup of $S_n$? $\endgroup$ – Rachid Atmai Nov 29 '19 at 23:33
  • $\begingroup$ Is that the only normal subgroup of $S_n$? Apparently so, yes, @16278263789, according to a Google search of "normal subgroups of $S_n$." $\endgroup$ – Shaun Nov 29 '19 at 23:36
  • $\begingroup$ Be careful though. It is the only proper normal subgroup, @16278263789. $\endgroup$ – Shaun Nov 29 '19 at 23:46
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    $\begingroup$ $A_n$ is the only proper non-identity normal subgroup of $S_n$, except that $S_4$ has an extra one, isomorphic to the Klein group. (The kernel of the action on the three partitions of shape $[2,2]$ of a set of size $4$.) $\endgroup$ – verret Nov 30 '19 at 0:23
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    $\begingroup$ Thank you to both of you, pretty cool. $\endgroup$ – Rachid Atmai Nov 30 '19 at 0:52

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