This isn't a homework problem as it is me working through some math texts in order to add some "rigor" to my physics/engineering acumen. As an undergrad, I attempted to take a course in abstract algebra, but had to drop it due to scheduling problems. I've been working through my copy of Hungerford and am using Tinkham's "Group Theory and Quantum Mechanics" to supplement the rather minimal treatment of group theory while providing some good practical problems. This problem is from the first chapter of Tinkham. 1. The problem statement, all variables and given/known data 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). 2. Relevant 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. 3. 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?