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### Lagrange's Theorem

```
Date: 01/24/2002 at 01:15:30
From: santhosh kumar.L
Subject: Modern algebra

Let G be a finite group of order p, where p is a prime number and G is
a cyclic group.

I need the proof of the theorem.
```

```
Date: 01/24/2002 at 10:09:27
From: Doctor Paul
Subject: Re: Modern algebra

Let |G| = p. Pick an element from G which is not the identity element
and call it x.

Compute:

x, x^2, x^3, ..., x^p = e

This list of p powers of x must include every element of G.

Restating, it must be the case that G = <x>. To see this, suppose
that this is not the case. Then <x> is some proper subgroup of G.
Lagrange's Theorem says that the order of this proper subgroup must
divide the order of G (which is the prime number p). Clearly this
cannot happen since the only divisors of the prime p are 1 and p.

Thus G = <x> and G is hence cyclic.

more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Modern Algebra

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