Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 3 / 5D / a / Solution

The order of an element $x$ is the least positive integer $d$ such that $x^d=e$, and the order of a finite group is its number of elements.

Lagrange theorem states that if $H\le G$ and $G$ is finite, thenIndeed, the left cosets of $H$ partition $G$. Multiplication by a coset representative is a bijection from $H$ to each coset, so every coset has $|H|$ elements. Summing over the $[G:H]$ cosets proves the formula.

The cyclic subgroup $⟨x⟩$ has exactly $d$ elements when $x$ has order $d$. Applying Lagrange to $⟨x⟩\le G$ gives

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