Extension splitting over Sylow subgroups

Question

Consider an extension of groups $$(E)\ :\ 1\to N\to G\stackrel{\pi}{\longrightarrow} Q\to 1$$ and assume

1. $N$ is abelian,
2. $Q$ is finite,
3. for any Sylow subgroup $S_p$ of $Q$, the pullback $(E_p)\ :\ 1\to N\to \pi^{-1}(S_p)\to S_p\to 1$ is split.

Then $(E)$ is split.

[A proof can be written using only elementary homological algebra plus Cartan-Eilenberg double coset formula.]

If we drop the first hypothesis, the conclusion doesn't hold in general. The simplest example seems to be the extension $$1\to F_2\to PSL_2(\mathbf Z)\simeq \mathbf Z/2\ast\mathbf Z/3\to PSL_2(\mathbf F_2)\simeq\mathfrak S_3\to 1$$

Even if we insist that $N$ be finite solvable, there are counter-examples (e.g. take the above with $PSL_2(\mathbf Z/8)$ instead of $PSL_2(\mathbf Z)$).

Assume $N$ is solvable and $\pi$ is split over all normalizers of Sylow subgroups in $Q$. Is $\pi$ necessarily split ?

Answer 1

This didn't seem likely, so I looked for a counterexample and found one after a short search. Clearly $Q$ must not have any normal Sylow subgroups, so the first example to try is $Q=S_4$. Then $N$ must not be abelian, so I tried $N=Q_8$ and found an example.

The example is $\mathtt{SmallGroup}(192,988)$ in GAP or Magma. This has seven generators $x_1,\ldots,x_7$, and the normal subgroup isomorphic to $Q_8$ is $\langle x_2, x_3x_7 \rangle$.

Checking that the extension is nonsplit but it splits over the normalizers of Sylow $2$- and $3$-subgroups is routine in GAP or Magma, but I can tell you the commands if it would help.

Magma code follows. Note that the Sylow $2$-subgroup is self-normalizing in $G$. The normalizer of the Sylow $3$-subgroup of $G$ contain $N$, so it is equal to the inverse image of the normalizer in $Q = G/N$ of a Sylow $3$-subgroup.

> G := SmallGroup(192,988);
> N := sub< G | G.2, G.3*G.7 >;
> #N;
8
> IsNormal(G,N);
true    
> #Complements(G,N);
0
> P := Sylow(G,2);
> P eq Normalizer(G,P);
true
> #Complements(P,N);
4
> NQ := Normalizer(G, Sylow(G,3));
> N subset NQ;
true
> #Complements(NQ,N);
1

Answer 2

In case anyone is interested, here is what happens for extensions $$(E)\ :\ 1\to H\to G\to \mathfrak S_4\to 1$$ when $H$ is $D_8$ or $Q_8$. There are 24 of these.

We begin with $D_8$. The set $F:=Hom(\mathfrak S_4, Out(D_8))$ has order two. For each $f\in F$, the set of extensions $E$ affording $f$ is non-empty, and thus is in bijection (up to equivalence) with $H^2(\mathfrak S_4,Z(D_8))\simeq \mathbf Z/2\times \mathbf Z/2)$. Thus there are $8$ equivalence classes of extensions $(E)$ with $H=D_8$.

These furnish $8$ non-isomorphic groups $G$ :

• SmallGroup(192,973) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,974) : the extension $E$ is split.
• SmallGroup(192,989) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,990) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,1472) : the extension $E$ is split.
• SmallGroup(192,1473) : the extension $E$ is split.
• SmallGroup(192,1485) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,1486) : the extension $E_{(2)}$ is non-split.

Similarly, there are 16 classes of extensions with $H=Q_8$, but these furnish only 12 isomorphism classes of G's :

• SmallGroup(192,975) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,976) : the extension $E_{}$ is split.
• SmallGroup(192,987) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,988) : the extension $E_{(2)}$ is split but $E$ isn't.
• SmallGroup(192,1477) : the extension $E_{}$ is split.
• SmallGroup(192,1478) : the extension $E_{}$ is split.
• SmallGroup(192,1483) : the extension $E_{(2)}$ is non-split.
• SmallGroup(192,1484) : the extension $E_{(2)}$ is non-split.

and some groups $G$ containing $2$ copies of $Q_8$ that are not exchanged by any automorphism of G :

• SmallGroup(192,1489) : in both cases the extension $E_{(2)}$ is non-split.
• SmallGroup(192,1490) : in both cases the extension $E_{}$ is split.
• SmallGroup(192,1492) : in both cases the extension $E_{(2)}$ is non-split.
• SmallGroup(192,1494) : in both cases the extension $E_{}$ is split.

So SmallGroup(192,988) is the unique smallest counter-example to the question (that it is a counter-example is explained in Derek Holt's answer).