Exercise 1700Algebra › GroupsOlympiadAdded 3 Nov 2026

Cauchy's theorem for groups

#Cauchy theorem#prime order#existence

Statement

Let $G$ be a finite group of order $n$ , and $p$ a prime dividing $n$ . Show that $G$ has an element of order $p$ .

Hint

Use induction on $|G|$ . If $G$ has a proper subgroup of order divisible by $p$ , conclude by induction. Otherwise, use the class equation.

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Solution

Induction on $|G|$ . If $G$ is abelian, Cauchy's theorem is classical (structure of finite abelian groups). In general, consider the class equation
$|G|=|Z(G)|+\sum_i[G:C(g_i)]$
where the sum is over non-central conjugacy classes.

Case 1: $p\mid|Z(G)|$ . Then $Z(G)$ has an element of order $p$ by the abelian case.

Case 2: $p\nmid|Z(G)|$ . Then some $[G:C(g_i)]$ is not divisible by $p$ , hence $p\mid|C(g_i)|$ with $|C(g_i)|<|G|$ . By induction on $C(g_i)$ , it has an element of order $p$ , hence so does $G$ .

Conclusion. $G$ has an element of order $p$ .

Cauchy's theorem for groups

Statement

Solution

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