 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-oxford/2015/prelims/mathematics-i/5/b/ii/solution

Choose

b_{0} \in B

and put

H = b_{0}^{- 1} B

. Then

e \in H

. For any

g_{1}, g_{2} \in G

g_{1} H = (g_{1} b_{0}^{- 1}) B, g_{2} H = (g_{2} b_{0}^{- 1}) B,

so these translates are equal or disjoint by the assumed property of

B

. Applying i shows that

H

B = b_{0} H, H = b_{0}^{- 1} B \leq G .

This proves that

B

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