Possible Duplicate:
Is Lagrange's theorem the most basic result in finite group theory?
I can't seem to find a simple proof of this in my textbook, not can I figure out a good way to search for it online.
Basically, I'm trying to show that $\forall g \in G$, $g^{\#(G)}=1$.
Obviously I can show this from Lagrange's theorem, but this requires introducing cosets to prove.
I'm wondering if there's some way to prove this from basic manipulations, perhaps relying on the additional properties of cyclic subgroups. Or is there no shortcut through this proof besides using Lagrange's theorem?
Note: This is not homework, but I'm actually looking to explain some basic group theory to someone else.