# Normal subgroups of $S_n$ and even permutations

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.

• Isn't that how the alternating group is defined? – verret Nov 29 '19 at 23:26
• 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? – Rachid Atmai Nov 29 '19 at 23:28

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 $$A_n$$.
• 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 $S_n$? – Rachid Atmai Nov 29 '19 at 23:33
• Is that the only normal subgroup of $S_n$? Apparently so, yes, @16278263789, according to a Google search of "normal subgroups of $S_n$." – Shaun Nov 29 '19 at 23:36
• $A_n$ 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 $4$.) – verret Nov 30 '19 at 0:23