 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2023/prelims/mathematics-i/6/b/iii/solution

k H

with

k \in K

is also a left coset of

H

G

. Moreover equality of two such cosets means the same thing in

K

and in

G

k H = k^{'} H ⟺ k^{' - 1} k \in H .

Thus the map from the set of left cosets of

H

K

to the set of left cosets of

H

G

, taking

k H

to itself, is injective. Consequently the index of a subgroup satisfies

∣ K / H ∣ \leq ∣ G / H ∣ \leq n .

These are sets of cosets; no normal subgroup assumption is needed.

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