# Simple groups that are maximal subgroups of G.

• Jan 16th 2012, 09:18 PM
ENRIQUESTEFANINI
Simple groups that are maximal subgroups of G.
Hi:
I'll make a straightforward question: Let $G$ be group and $S \leq T \leq G$, these groups satisfying the following conditions:

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

Is there any other piece of information that can be deduced from these three facts? In particular, can I infer $S \unlhd G$?
• Jan 17th 2012, 01:03 AM
Swlabr
Re: Simple groups that are maximal subgroups of G.
Quote:

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

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

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

Well, no, because $T$ is simple. If $S\leq T\leq G$ and $S\lhd G$ then $S\lhd T$...
• Jan 17th 2012, 01:40 AM
ENRIQUESTEFANINI
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.