= Solution
{c}
<Lagrange's theorem> states that if $H$ is a <subgroup> of a <finite group> $G$, then
$$
\boxed{|G|=[G:H]|H|,\qquad |H|\mid|G|,}
$$
where the <index of a subgroup> $[G:H]$ is the number of distinct <left cosets>.
Every $g\in G$ lies in $gH$, since $g=ge$. By <5/a/i>, distinct <left cosets> are disjoint, so these <sets> form a <partition of a set> $G$. Each has <cardinality> $|H|$: the <function> $h\mapsto gh$ from $H$ to $gH$ is a <bijection>, with inverse $gh\mapsto h$, by cancellation in a <group>. Counting the elements in this finite <partition of a set> gives $|G|=[G:H]|H|$, proving the theorem.
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