Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I have to find a group $G$ and a subgroup $H$ with $[G:H]=7$ such that for every $N$ normal subgroup of $G$ with $N\subset H$ we have $[G:N]\geq 7!$, could you help me please?

share|cite|improve this question
The simplest way this could happen is if there are no nontrivial normal subgroups contained in $H$, and $G$ has order at least $7!$. Can you think of a group of order $7!$ with few normal subgroups? – Chris Eagle Jan 3 '12 at 2:04
Hint. If $N$ is normal in $G$ and contained in $H$, then it is also normal in $H$. Can you think of a group $H$ with the property that if $N\triangleleft H$, $N\neq H$, then $[H:N]\geq 6!$? – Arturo Magidin Jan 3 '12 at 2:05

Pick a simple group $S$ of order larger that $7!$ and let $G=C_7\times S$ be the direct product of $S$ and a cyclic group of order $7$. Pick $H$ to be $S\subseteq G$, which has index $7$ in $G$; there are no non-trivial normal subgroups of $G$ contained in $H$, so your second condition more or less vacuously holds.

share|cite|improve this answer
In fact, you just need $|S|\geq 6!$. If you are allowed to take $N=H$ (unclear in the original post), then a semidirect product and a different $H$ will do... – Arturo Magidin Jan 3 '12 at 2:13
could you tell me how can you solve the problem if we are allowed to take N=H, please? – Alex M Jan 3 '12 at 2:25
@Alex: Can you think of a group $G$ of order $7!$ with very few normal subgroups, that has a subgroup of $H$ index $7$, with $H$ having very few normal subgroups, no nontrivial one of which is normal in $G$? It's a very standard group. – Arturo Magidin Jan 3 '12 at 6:07

Your Answer


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

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