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The strong Cayley theorem states that if $G$ is a group, $U$ is a subgroup of $G$ with $(G:U)=m$, then there is a homomorphism $\varphi: G \longrightarrow S_m$, whose kernel is the largest normal subgroup of $G$ which is contained in $U$.

With this theorem one application is e.g. finding non-trivial normal subgroups of a group $G$. I already got results like if $|G| \nmid m!$, then $G$ is not simple and the one which states that a subgroup with index $p$, where $p$ is the smallest prime divisor of a group, is a normal subgroup.

Are there more (bigger) theorems which one can follow from strong Cayley (maybe concerning simplicity)?

Arnaud D.
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  • I'd say that this construction (permutation representation on cosets of a subgrop) is one of main tools in theory of permutation groups (which is, roughly speaking, taxonomy of homomorphisms to symmetric groups). For example, it's a way to prove that every 4-transitive https://groupprops.subwiki.org/wiki/Multiply_transitive_group_action group is simple. – xsnl Mar 30 '18 at 18:38

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A "bigger" application for the strong Cayley Theorem is another proof for the fact that any simple group of order $60$ is isomorphic to the group $A_5$:

Any simple group of order $60$ is isomorphic to $A_5$

Dietrich Burde
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