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Let $G$ be a centerless finitee group and $\pi(G)=\{p,q,r\}$ be the set of prime divisors of $|G|$. I have some questions about these groups.

  1. Is there any classification of such groups?
  2. In what conditions, we can say that $G$ is solvable?
  3. In what conditions, we can say that $G$ has a non-trivial normal abelian subgroup?

My main question is about arbitrary primes p,q and r. If the answer of the above questions are not known in general, one can answer my questions for centerless $\{2,3,5\}$-groups.

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    $\begingroup$ The solubility is already discussed in Herzog (1968). Accordingly, there exists a non-soluble group of order $p^aq^br^c$ if and only if $(p,q,r)=(2,3,5)$ with $a\geq2$, $(p,q,r)=(2,3,7)$ with $a\geq2$, $(p,q,r)=(2,3,13)$ with $a\geq4$ and $b\geq3$, or $(p,q,r)=(2,3,17)$ with $a\geq4$ and $b\geq2$. $\endgroup$ – M. Farrokhi D. G. Nov 22 '15 at 6:17
  • $\begingroup$ Dear Farrokhi, thank you for your answer. In the Herzog's paper only finite simple groups are discussed. My question is about arbitrary groups with trivial centers. $\endgroup$ – majid arezoomand Nov 22 '15 at 19:24
  • $\begingroup$ Yes, but every non-soluble group has a section that is a minimal non-abelian simple group. $\endgroup$ – M. Farrokhi D. G. Nov 24 '15 at 1:23

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