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About normal minimal subgroups not in the Frattini

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

  1. Tom1909

    Tom1909 Guest

    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?

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