Question

Question: Use Cauchy’s Theorem to prove that a finite p-group has order${\mathit{p}}^{\mathbf{n}}$ for some$\mathit{n}\mathbf{\ge }\mathbf{0}$

Short answer

It is proved that a finite p-group has order$p^n$ for some$n\ge 0$

Step-by-step solution

Cauchy’s Theorem

If G is a finite group whose order is divisible by a prime p, then G contains an element of order p.

Proving that a finite p-group has order forpn some n≥0

Suppose G is a finite p-group, and all element of G has some power of p.

Let’s assume the order of G is not the power of p.

This implies that there is another prime qthat is not equal to p and divides$\left|G\right|$ .

Therefore, according to Cauchy’s Theorem, G has some element whose order is q.

This is contradicting our assumption because we assumed that G is a finite p-group, and all elements of G has some power of p.

Therefore, our assumption is wrong.

Hence, it is proved that a finite p-group has order $p^n$for some$n\ge 0$