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Leyli Jafari
Leyli Jafari
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Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, $H'$ is core-free, then is it true that $G$ is simple (or equivalently $\Phi(G)$ necessarily is trivial)?

Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, $H'$ is core-free, then is it true that $\Phi(G)$ necessarily is trivial?

Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, $H'$ is core-free, then is it true that $G$ is simple (or equivalently $\Phi(G)$ is trivial)?

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Leyli Jafari
Leyli Jafari
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Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, either $H'$ is core-free or $H' \leq \Phi(G)$, then is it true that $\Phi(G)$ necessarily is trivial?

Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, either $H'$ is core-free or $H' \leq \Phi(G)$, then is it true that $\Phi(G)$ necessarily is trivial?

Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, $H'$ is core-free, then is it true that $\Phi(G)$ necessarily is trivial?

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Leyli Jafari
Leyli Jafari
  • 347
  • 4
  • 13

Minimal non-solvable groups with a special property

Suppose that $G$ is a finite minimal non-solvable group (i.e. a non-solvable group all of whose proper subgroups are solvable) whose Frattini subgroup $\Phi(G)$ is abelian. If for every subgroup $H < G$, either $H'$ is core-free or $H' \leq \Phi(G)$, then is it true that $\Phi(G)$ necessarily is trivial?