- Prove Cauchy’s theorem.
Induce on \(\size G\). Assume \(\size G > p\) and pick \(g\neq 1\). If \(p\divides \size g\), use cyclic group theory, so assume otherwise. Use that \(\size G = \size G/N \size N\) so \(p\) divides \(\size G/N\), apply IH to get an element of order \(p\) in the quotient. Then \(y\not\in N\) but \(y^p\in N\), so \(\gens{y}\neq \gens{y^p}\) since \(y^p\in N \implies \gens{y^p} \subseteq N\). Get \(p\divides \size \gens{y}\), apply IH.
Solution
Cauchy’s theorem: if \(G\) is a finite group and \(p\) is a prime dividing \(|G|\), then \(G\) has an element of order \(p\).
Let \(X=\{(g_1,\dots,g_p)\in G^p: g_1g_2\cdots g_p=1\}\), and let \(C_p=\langle\sigma\rangle\) act on \(X\) by \(\sigma(g_1,\dots,g_p)=(g_2,\dots,g_p,g_1)\).
1. \(|X|=|G|^{p-1}\), so \(p\mid|X|\).
\(g_1,\dots,g_{p-1}\) are arbitrary and determine \(g_p=(g_1\cdots g_{p-1})^{-1}\). The cyclic shift preserves \(X\), since \(g_2\cdots g_pg_1=g_1^{-1}(g_1\cdots g_p)g_1=1\).
2. Every \(C_p\)-orbit in \(X\) has size \(1\) or \(p\), and the orbits of size \(1\) are the tuples \((g,\dots,g)\) with \(g^p=1\).
Orbit sizes divide \(|C_p|=p\). A tuple is fixed by \(\sigma\) exactly when all its entries are equal, and \((g,\dots,g)\in X\) exactly when \(g^p=1\).
3. Q.E.D.
Let \(F\) be the set of fixed tuples. By steps 1 and 2, \(|F|\equiv|X|\equiv0\pmod p\), and \((1,\dots,1)\in F\), so \(|F|\ge p\ge2\). Hence some \((g,\dots,g)\in F\) has \(g\neq1\) and \(g^p=1\), and \(g\) has order \(p\).