Linked Questions

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Herstein exercise: A subgroup of a finite group G such that $|G| \nmid i_G(H)!$ must contain a non-trivial normal subgroup.

This is a 'Harder' problem 40 from Abstract Algebra(1996) by Herstein. I'm just not able to figure out how to do this. even though I found a very similar post. Following is a verbatim statement of the ...
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1answer
82 views

group theory simple group subgroup index |G| < n!

Show that if $G$ is a simple group with a subgroup $H$ of index $n>1$, then $|G| \leq n!$. Hence show that a group of order $2^k \times 3$ can never be simple for $k>1$. So I have let $X$ be ...
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$G$ be a finite group, if $H\lneq G$ and $|G| \nmid [G:H]!$. Prove $G$ is not simple [duplicate]

Let G be a finite group, if $H\lneq G$ and $|G| \nmid ( \, [G:H]! \, ) $ then prove $G$ is not simple. I used contrapositive argument. Suppose $G$ is simple then we need to prove that $ |G| \mid ( \,...

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