Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2024 / Paper 2 / 8 / c / Solution

Let $d=\mathrm{ord}(g)$. The powersare distinct: equality of two with exponents between $0$ and $d-1$ would give a smaller positive power equal to $e$. Reducing exponents modulo $d$ proves closure and supplies inverses, so this is a cyclic subgroup of group order $d$. Lagrange's theorem now gives