Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics I / 6 / b / iv

Solution

Φ : G / (H \cap K) ⟶ (G / H) \times (G / K), g (H \cap K) ⟼ (g H, g K) .

It is well defined: replacing

g

gu

for

u \in H \cap K

changes neither component. It is injective because

g H = g^{'} H and g K = g^{'} K ⟺ g^{' - 1} g \in H \cap K ⟺ g (H \cap K) = g^{'} (H \cap K) .

The Cartesian product on the right has at most

nm

elements. Therefore

∣ G / (H \cap K) ∣ \leq ∣ G / H ∣ ∣ G / K ∣ \leq nm .

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