- #1

- 8

- 0

## Homework Statement

Consider a group of order (p-1), where p is a prime, with elements 1,2,..,(p-1) and group multiplcation is defined as multiplcation mod p. Prove that for any element of the group, A, the following relationship holds: A^(p-1)=E (the multiplicative identity).

## Homework Equations

This is equivalent to proving that for a given integer, N, and a prime, P, that N^P is congruent to N modulo P.

## The Attempt at a Solution

The solution I have is very simple, but I feel as if it is missing something

-Assume that any element, A, generates a cyclic subgroup of order k, s.t. A^k=E.

-By Lagrange's theorem, k must be an integer divisor of (p-1), the order of the larger group. This leads to n*k=(p-1).

-A^(p-1)=A*(n*k)=(A^k)^n=(E)^n=E, Q.E.D.

I like the proof, but the first statement doesn't seem complete. I don't feel comfortable just assuming that

*any*element

*can*generate such a subgroup. Can anyone help me justify this?