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?
Subgroup of index 7
0
$\begingroup$
group-theory
finite-groups
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1The 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? – 2012-01-03
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1**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 $[H:N]\geq 6!$? – 2012-01-03