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Alex M.
Alex M.
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I have aTheorem 2 in my preprint https://arxiv.org/abs/math/0312342"Group Orders That Imply Existence of Nontrivial Normal $p$-Subgroups" with divisibility conditions on $n$shows that guarantee existence of a nontrivialif $p$-subgroup in any group of order$|G| = p^s m$ and $n$$p \nmid \Gamma(m)$ (defined therein), then either $O_p (G) \ne 1$, or $G$ is not solvable.

I have a preprint https://arxiv.org/abs/math/0312342 with divisibility conditions on $n$ that guarantee existence of a nontrivial $p$-subgroup in any group of order $n$.

Theorem 2 in my preprint "Group Orders That Imply Existence of Nontrivial Normal $p$-Subgroups" shows that if $|G| = p^s m$ and $p \nmid \Gamma(m)$ (defined therein), then either $O_p (G) \ne 1$, or $G$ is not solvable.

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rvf0068
rvf0068
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I have a preprint https://arxiv.org/abs/math/0312342 with divisibility conditions on $n$ that guarantee existence of a nontrivial $p$-subgroup in any group of order $n$.