Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 5 / b / ii

Table of contents
- Solution 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

is a subgroup. Thus

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

This proves that

B

is a left coset. Together with i, it gives the equal-or-disjoint translates characterize cosets criterion.

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