Schur-Zassenhaus Theorem

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Theorem

Let $G$ be a finite group and $N$ be a normal subgroup in $G$.

Let $N$ be a Hall subgroup of $G$.


Then there exists $H$, a complement of $N$, such that $G$ is the semidirect product of $N$ and $H$.


Proof

The proof proceeds by induction.

By definition, $N$ is a Hall subgroup if and only if the index and order of $N$ in $G$ are relatively prime numbers.


Let $G$ be a group whose identity is $e$.

We induct on $\order G$, where $\order G$ is the order of $G$.

We may assume that $N \ne \set e$.

Let $p$ be a prime number dividing $\order N$.

Let $\Syl p N$ be the set of Sylow $p$-subgroups of $N$.

By the First Sylow Theorem:

$\Syl p N \ne \O$

Let:

$P \in \Syl p N$
$G_0$ be the normalizer in $G$ of $P$
$N_0 = N \cap G_0$.

By Frattini's Argument:

$G = G_0 N$

By the Second Isomorphism Theorem for Groups and thence Lagrange's Theorem (Group Theory), it follows that:

$N_0$ is a Hall subgroup of $G_0$
$\index {G_0} {N_0} = \index G H$

Suppose $G_0 < G$.

Then by induction applied to $N_0$ in $G_0$, we find that $G_0$ contains a complement $H \in N_0$.

We have that:

$\order H = \index {G_0} {N_0}$

and so $H$ is also a complement to $N$ in $G$.

So we may assume that $P$ is normal in $G$ (that is: $G_0 < G$).



Let $Z \paren P$ be the center of $P$.

By:

Center is Characteristic Subgroup
$P$ is normal in $G$
Characteristic Subgroup of Normal Subgroup is Normal

$Z \paren P$ is also normal in $G$.

Let $Z \paren P = N$.

Then there exists a long exact sequence of cohomology groups:

$0 \to H^1 \paren {G / N, P^N} \to H^1 \paren {G, P} \to H^1 \paren {N, P} \to H^2 \paren {G / N, P} \to H^2 \paren {G, P}$

which splits as desired.





Otherwise:

$Z \paren P \ne N$

In this case $N / Z \paren P$ is a normal (Hall) subgroup of $G / Z \paren P$.

By induction:

$N / Z \paren P$ has a complement $H / Z \paren P$ in $E // Z \paren P$.




Let $G_1$ be the preimage of $H // Z \paren P$ in $G$ (under the equivalence relation).



Then:

$\order {G_1} = \order {K / Z \paren P} \times \order {Z \paren P} = \order {G / N} \times \order {Z \paren P}$



Therefore, $Z \paren P$ is normal Hall subgroup of $G_1$.


By induction, $Z \paren P$ has a complement in $G_1$ and is also a complement of $N$ in $G$.



$\blacksquare$


Also known as

Some sources refer to this theorem as Schur's theorem, but that name is also used for an unrelated result in Ramsey theory.


Source of Name

This entry was named for Issai Schur and Hans Julius Zassenhaus.