Minimal non solvable group is simple

Question

I suppose to prove in my homework that

Every minimal non solvable group is simple.

I can't find the way. Thank you.

Answer 1

If $N$ is a proper non-trivial normal subgroup of $G$, then both $G/N$ and $N$ are solvable, whence $G$ is solvable, a contradiction.

Answer 2

Sometimes a minimal non-$\mathfrak{X}$ group is defined as a group which is not $\mathfrak{X}$, but all of whose proper subgroups are $\mathfrak{X}$. (Here $\mathfrak{X}$ is a group-theoretic property like solubility.)

With this interpretation, the result is false. Take $G = \operatorname{SL}(2,5)$. Here $Z(G)$ has $2$ elements, and $G/Z(G) \cong A_{5}$. One can verify that all proper subgroups of $G$ are soluble, the important step being possibly that $G$ does not split over $Z(G)$. This depends on the fact that the generator $$ \begin{bmatrix}-1&0\\0&-1\end{bmatrix} $$ of $Z(G)$ is the square of $$ \begin{bmatrix}0&2\\2&0\end{bmatrix}. $$

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If you include in the definition the requirement that proper quotients are also in $\mathfrak{X}$, then Nicky Hekster's proof applies.