# About normal minimal subgroups not in the Frattini

Discussion in 'Mathematics' started by Tom1909, Oct 8, 2018.

1. ### Tom1909Guest

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?