1
$\begingroup$

Let $\mathbb{P}$ is a set primes numbers, $\pi \subseteq P$ and $\pi ^{\prime }=P-\pi $

Let $G$ abelian and $O_{\pi }\left( G\right) =\left\langle N~;~N\trianglelefteq G\text{ and }% N\text{ is }\pi \text{-subgroup}\right\rangle $

Is it true that $O_{\pi^{\prime } }\left( G\right) =1\Longrightarrow G$ is $\pi -$group?

I think so, because every subgroup is normal in $G$.

  • 0
    Yes, this is correct assuming $G$ is finite (or at least torsion), for exactly the reason you say (and Sylow's theorem).2012-08-15
  • 0
    Lima: Thank you! I am needing that result.2012-08-15
  • 1
    You can relax your condition of G being abelian to G being *$\pi$-separable* (that is G has a normal series with each factor either a $\pi$- or $\pi$'-group.2012-08-15

0 Answers 0