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Past exam of the mathematics course of the University of Oxford

2023

Prelims

Mathematics I

{c}

Define a map between sets of <left cosets> by
$$
\Phi:G/(H\cap K)\longrightarrow(G/H)\times(G/K),
\qquad g(H\cap K)\longmapsto(gH,gK).
$$
It is well defined: replacing $g$ by $gu$ for $u\in H\cap K$ changes neither component. It is injective because
$$
gH=g'H\text{ and }gK=g'K
\iff g'^{-1}g\in H\cap K
\iff g(H\cap K)=g'(H\cap K).
$$
The <Cartesian product> on the right has at most $nm$ elements. Therefore
$$
\boxed{|G/(H\cap K)|\le|G/H|\,|G/K|\le nm.}
$$
This proves the basic bound for a <finite intersection of finite-index subgroups>.

Solution

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

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

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