Normal subgroups of $S_n$ and even permutations

Question

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.

Answer 1

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 $A_n$.