 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2021/prelims/mathematics-i/6/a/ii/solution

For

W = a H

and

W^{'} = b H

f : W \to W^{'}, f (w) = b a^{- 1} w .

w = ah

then

f (w) = bh \in b H

, so this function is well defined. Its inverse function is

w^{'} \mapsto a b^{- 1} w^{'}

, since successive left translation in a group by

b a^{- 1}

and

a b^{- 1}

cancels. Thus

f : a H \sim b H is a bijection .

In particular every left coset has the same cardinality as

H

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