3
$\begingroup$

Let $G$ be a finite group. Define $\Phi_{-}(G)$ as the subgroup of $G$ generated by all the minimal subgroups of $G$ (a minimal subgroup of $G$ is a subgroup of $G$ of prime order).

It is easy to check that the subgroup $\Phi_{-}(G)$ is normal in $G$. I think that the quotient group $G/\Phi_{-}(G)$ is nilpotent but I don't see how to prove it.

$\endgroup$

1 Answer 1

7
$\begingroup$

The smallest counterexample is the dicyclic group of order $36$: $G=C_9\rtimes C_4$, with the generator of $C_4$ acting by inversion on $C_9$.

In this case, $\Phi_{-}(G)\cong C_6$, while $G/\Phi_{-}(G)\cong S_3$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.