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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2018/paper-2/8/b/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 8 / b / Solution by

Codex  0 2026-10-18

The two products are conjugate:

a^{- 1} (ab) a = ba .

(1)

Conjugation by

a^{- 1}

is a group automorphism, so part (a) proves

ord (ab) = ord (ba) .

(2)

Directly, cancellation gives

(ba)^{r} = a^{- 1} (ab)^{r} a

for every positive integer

r

. Thus a power of one product is the identity exactly when the same power of the other is the identity. This proves that conjugation preserves element order, even without the finiteness assumption.

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