 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2021/ia/paper-3/5d/d/ii/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

Among the

n + 1

right cosets

H, H x, H x^{2}, \dots, H x^{n}

two must coincide because

[G : H] = n

. If

H x^{i} = H x^{j}

with

0 \leq i < j \leq n

x^{- i}

gives

H = H x^{j - i}

, so

x^{j - i} \in H, 0 < j - i \leq n .

The least such exponent need not divide

n

. For example, take

G = S_{3}

H = ⟨(12)⟩

, and

x = (13)

. Then

[G : H] = 3

, while

x \in / H

and

x^{2} = e \in H

, so the least exponent is two, which does not divide three.

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