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Problem 86 (Lagrange's Theorem Proof)

Prove Lagrange's Theorem.

Theorem (Lagrange's Theorem)

Suppose that $G$ is a finite group and that $H$ is a subgroup of $G$. Then the order of $H$ divides the order of $G$. In particular, we know that $|G|/|H|$ equals the number of distinct right (or left) cosets of $H$ in $G$.

Solution

Let $G$ be a finite group with order $g$. Let $H$ be a subgroup of $G$ with order $h$. Thus $h\leq g$. We must show that $h$ divides $g$.

Pick $a\in G$. Thus we have $|H|=|Ha|$.

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