TITLE: About normal minimal subgroups not in the Frattini
QUESTION [9 upvotes]: In Neukirch--Schmidt--Wingberg, "Cohomology of Number Fields", Second edition, page 624, Exercise 2, it is stated the following fact. 
$\textbf{Claim}$: If $N$ is a normal subgroup, minimal among normal subgroups, of a group $G$ not contained in the Frattini's subgroup of $G$, then $G$ is a semi-direct product over $N$, i.e. there exists $H$ subgroup of $G$ with $H \cap N=\{id\}$ and $HN=G$. 
Now, if $N$ is a simple non-abelian normal subgroup of $G$, the above Claim implies that $G$ is a semi-direct product over $N$. Indeed the Frattini of $G$ is nilpotent, and since $N$ is non-abelian, it cannot be contained in a nilpotent group, otherwise $N$ itself would be nilpotent, and nilpotent and simple implies having prime order and hence abelian. Moreover $N$ is clearly minimal, since $N$ itself does not have non-trivial proper normal subgroup. But then I see a clear contradiction with Lemma 4.3 of this paper of Lucchini, Menegazzo and Morigi https://projecteuclid.org/download/pdf_1/euclid.ijm/1258488162.
Could somebody explain (in case I did not perform a mistake in reasoning)  which of the two publications has a mistake?

REPLY [13 votes]: The first claim is false: We will give an elementary proof using GAP. Consider the following code:
U := SmallGroup(720, 765);
N := Group([ (1,2,9,3)(4,6,10,8), (3,4,9,10)(5,8,6,7) ]);
P := SylowSubgroup(U, 2);

Running isSimple(N) verifies that N is a simple group, and running IsNormal(U,N) verifies that N is normal in U. We claim that U does not split over N: The subgroup N has index 2 and therefore if it had a complement, that would mean there is a cyclic subgroup of size 2 that is disjoint from N. This is impossible. Indeed, Zuppos(P) gives a list of all cyclic subgroups of P and every cyclic subgroup of size 2 is in N. Here is some quick code to check this (prints all true):
list := Zuppos(P);
for i in list do 
if Order(i) = 2 then Print(IsSubgroup(N,Group([i])));
fi; od;

Hence U does not split over N, as N is normal and all 2-Sylow subgroups are conjugate. (In fact, one can run the above code over all of U and use almost no theory whatsoever.)
Finally, N is not contained in the Frattini since it is not inside the maximal subgroup of U containing P, since P and N collectively generated U by index considerations.
PS: U never splits, as N is the only non-trivial normal subgroup of U (just run NormalSubgroups(U)). The group U is the Mathieu group of degree 10 and N is the alternating group on 6 letters. Moreover, P is the Semidihedral group SD16.