Every prime dividing a finite group order occurs as an element order.Suggest a slogan
Let \(G\) be a finite group and let \(p\) be prime. If \(p\) divides \(\abs G\), then \(G\) contains an element of order \(p\), and therefore a subgroup of order \(p\).
By Lagrange's theorem, the order of every element of \(G\) divides \(\abs G\); Cauchy’s theorem is the converse for prime divisors of \(\abs G\). Sylow's first theorem strengthens the existence statement from order \(p\) to order \(p^a\), where \(p^a\) is the largest power of \(p\) dividing \(\abs G\).