# Thread: Simple groups that are maximal subgroups of G.

1. ## Simple groups that are maximal subgroups of G.

Hi:
I'll make a straightforward question: Let $\displaystyle G$ be group and $\displaystyle S \leq T \leq G$, these groups satisfying the following conditions:

(a) $\displaystyle T \unlhd G$.
(b) $\displaystyle T$ is simple.
(c) $\displaystyle T$ is a maximal subgroup of$\displaystyle G$.

Is there any other piece of information that can be deduced from these three facts? In particular, can I infer $\displaystyle S \unlhd G$?

2. ## Re: Simple groups that are maximal subgroups of G.

Originally Posted by ENRIQUESTEFANINI
Hi:
I'll make a straightforward question: Let $\displaystyle G$ be group and $\displaystyle S \leq T \leq G$, these groups satisfying the following conditions:

(a) $\displaystyle T \unlhd G$.
(b) $\displaystyle T$ is simple.
(c) $\displaystyle T$ is a maximal subgroup of$\displaystyle G$.

Is there any other piece of information that can be deduced from these three facts? In particular, can I infer $\displaystyle S \unlhd G$?
Well, no, because $\displaystyle T$ is simple. If $\displaystyle S\leq T\leq G$ and $\displaystyle S\lhd G$ then $\displaystyle S\lhd T$...

3. ## Re: Simple groups that are maximal subgroups of G.

That's a pity, because I'm in the process of solving a problem, half of which, it seems, I have already solved, and I thought I could extract aditional info from those facts. About the problem it would be unfair to speak because I've already posted in MHF stating its hypothesis (and the propositions I have arrived at up to now). Thanks for your reply.