A note on the Schur-Zassenhaus theorem

I haven’t posted in a while; it’s time to dust off this blog and get things started again with a guest post!

In this post, guest user Anon1 posted a proof of the existence of finite fields of every possible order. Today I’ll share a writeup by the same user on the Schur-Zassenhaus Theorem in group theory!

The theorem statement

Let $G$ be a finite group, and let $N$ be a normal subgroup of $G$ such that $\gcd(|N|,[G:N])=1$. Then the Schur-Zassenhaus theorem says that:

$G$ has a subgroup $H$ such that $H \cong G/N$.
Any two such subgroups $H_1,H_2$ of $G$ are conjugate in $G$.

Recall that a subgroup $N$ is normal if $gNg^{-1}=N$ for all $g\in G$, and the quotient $G/N$ is the group of cosets $gN$.

An example

Let’s look at an example. In the symmetric group $S_3$, the alternating subgroup $A_3$ of odd permutations consists of the three cycles $(123)$ and $(132)$ along with the identity element, and is isomorphic to the cyclic group $C_3$. It has index $2$ in $S_3$, and $|A_3|=3$, and $2$ and $3$ are relatively prime, so the Schur-Zassenhaus theorem applies!

So $S_3$ should have a subgroup isomorphic to $C_2$, and indeed it does - it has three such subgroups in fact, those generated by the transpositions $(12)$, $(13)$, or $(23)$. These are indeed all conjugate to each other as well, by simply renaming the elements.

A nonexample

Can we do the same construction for $S_4$, or higher symmetric groups? The alternating group $A_4$ is still a normal subgroup, but now $|A_4|=12$ which is not relatively prime to $2$, the size of the quotient. So Schur-Zassenhaus does not apply. And indeed, not all of the subgroups of $S_4$ isomorphic to the quotient $C_2$ are conjugate to each other: The subgroup generated by $(12)$ is not conjugate to the subgroup generated by $(12)(34)$, since conjugation by permutations simply renames the letters in cycle notation!

The writeup and proof

User Anon1 sent me a full writeup of a proof of the first statement using affine geometry over fields of prime order, and it’s available here as a PDF at this link. Enjoy!