1. This site uses cookies. By continuing to use this site, you are agreeing to our use of cookies. Learn More.

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?

    Login To add answer/comment

Share This Page