Question

Suppose that G is a group with subgroups K ≤ H ≤ G. Suppose that K...

Suppose that G is a group with subgroups K ≤ H ≤ G. Suppose that K is normal in G. Let G act on G/H, the set of left cosets of H, by left multiplication. Prove that if k ∈ K, then left multiplication of G/H by k is the identity permutation on G/H.

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Answer #1

Let K≤H≤G (where ≤ means subgroup). Let I be an index set for the set of left cosets of H in G, i.e.

|I |= [G : H ] and let J be an index set for the set of left cosets of K in H, i.e.: |J | = [ H:K ] . Since H≤G and K≤H , we have that

and

It remains to show that (gi hj )K (gt hs)K= ∅ for i,t ∈I , j,s ∈J and i ≠ t or j≠ s . But gi H gt H = ∅ for ∀i≠t ⇒ gi (hj K) gt (hsK)= ∅.

Likewise, since

hj K hsK= ∅ ⇒ gi (hj K) gi (hsK)= ∅. ∀j≠s . Thus, [G :K ] =|IJ|=[G :H ] [H:K]

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