Proving the bound $[G:Core(H)]\leq [G:H]!$

Question

I was having trouble figuring out how exactly to prove $[G:Core(H)]\leq [G:H]!$ where G is a group and H is a subgroup of G.

I know Core is the kernel of the homomorphism from G to $S_{G/H}$ induced by left multiplication on cosets, and it seems pretty obvious that the inequality should hold but I'm struggling on how to approach the proof.

Answer 1

You're very close to the answer.

Let $\rho:G\to S_{G/H}$ be the homomorphism you describe. Then

$$[G:Core_G(H)]=|G|/|ker(\rho)|=|\rho(G)|\le|S_{G/H}|=|G/H|!=[G:H]!$$