Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 7D / Solution

An action is a map $G\times X\to X$ satisfying $ex=x$ and $(gh)x=g(hx)$; it induces $G\to S_X$. Acting on the $n$ left cosets of $H$ gives a homomorphism with kernel $K◃G$ contained in $H$. If $K$ were trivial, $|G|$ would divide $n!$, so under the stated hypothesis $K$ is nontrivial. For $[G:L]=p$ with $p$ the least prime divisor of $|G|$, the transitive image in $S_p$ has order divisible by $p$; its other possible prime divisors are smaller, hence absent, so its order is $p$. Thus the kernel has index $p$ and, being contained in $L$, equals $L$. Prime index alone is insufficient: a subgroup generated by a transposition in $S_3$ has index $3$ and is not normal.

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