Every prime divisor of a finite group order occurs as an element order.Suggest a slogan
Let \(G\) be a finite group and \(p\) a prime dividing \(\abs{G}\). Then \(G\) has an element of order \(p\), and hence a subgroup of order \(p\).
By Lagrange's theorem, the order of every element of \(G\) divides \(\abs G\); conversely, Cauchy’s theorem gives an element of order \(p\) for every prime \(p\) dividing \(\abs G\). Sylow's first theorem strengthens it: if \(p^a\) is the largest power of \(p\) dividing \(\abs G\), then \(G\) has a subgroup of order \(p^a\).