Cauchy's theorem is the first partial converse to Lagrange's theorem: whenever a prime \(p\) divides the order of a finite group, that prime occurs as the order of an element. [@isaacs2008]
Definition
The hypothesis is \(p\mid|G|\), and the conclusion is equivalently the existence of a subgroup \(C_p\leq G\).
Core result
The result follows immediately from Sylow existence. McKay's independent counting proof lets \(C_p\) cyclically rotate the \(p\)-tuples satisfying \(x_1\cdots x_p=1\). Nonconstant orbits have size \(p\), so fixed tuples are congruent to the total modulo \(p\), yielding a nonidentity \(x\) with \(x^p=1\). See Problem 1A.8.
Structural properties
- Applying the theorem repeatedly in quotients and centers helps construct normal series in \(p\)-groups.
- Together with Lagrange, it says that the prime element orders of a finite group are exactly the prime divisors of its order.
- If \(p^a\mid|G|\), Cauchy gives only an element of order \(p\); a subgroup of order \(p^a\) requires the Sylow theorem.
- For abelian groups one may use power homomorphisms and induction; the nonabelian case needs a more general action or counting argument.
- The theorem does not guarantee an element of order \(p^2\): every nonidentity element of \(C_p\times C_p\) has order \(p\).
The source presents both the short deduction from Sylow existence and an independent exercise-derived action proof, illustrating that the same existence theorem can arise at different levels of machinery.
Example and boundary
Knowledge network
Sylow theory supplies the stronger prime-power subgroup statement, while the class equation and p-group articles show how Cauchy elements enter centers and normal series.
Proof and sources
Isaacs, Theorem 1.9 deduces Cauchy from Sylow existence; Problem 1A.8 contains the independent cyclic-action proof extracted here. [@isaacs2008]
