For which integers $n \ge 3$ is the dihedral group $D_{2n}$ a subgroup of $Alt_n$
Disclaimer, we are supposed to get the answer without Lagrange's Theorem.
I have just started working with the Alternating group $Alt_n$, or also denoted $\mathbb{A}_n$.
$$Alt_n := \{ f\in S_n \mid \operatorname{sg}(f)=1 \}$$
$\operatorname{sg}(f)=1$ when f is a product even pair of 2-cycles in the symmetric group $S_n$, which equivalently means that f is product of 3-cycles.
So this is all very new to me, and I am at a loss of how to consider how the dihedral group could be a subgroup of $Alt_n$. Define:
$$D_{2n} = \langle r,s\mid r^n = s^2 = 1, sr^i=r^{-i}s\rangle$$
In this part of the class we are talking about sub groups generated by sub sets of the group; order of subgroups; the subgroup criterion, which says for a subset to be a subgroup then it need to be closed under product and inverses. So I assume the first step is to show that $D_{2n}$ is a sub set of $S_n$. But I'm just not making the connection.
It seems like elements of $S_n$ will always have an odd order, so perhaps $n$ needs to be odd.
So like I said, I am a little at a loss of even the criteria that we would need to show that $D_{2n}$ is a sub group of $Alt_n$. Thanks for the help!