Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $G$.

share|improve this question
    
Yes, this is correct assuming $G$ is finite (or at least torsion), for exactly the reason you say (and Sylow's theorem). –  Jack Schmidt Aug 15 '12 at 19:39
    
Lima: Thank you! I am needing that result. –  User2040 Aug 15 '12 at 20:09
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. –  Nicky Hekster Aug 15 '12 at 21:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.