# Any group of 3 elements is isomorphic to Z3

by kathrynag
Tags: elements, isomorphic
 P: 603 1. The problem statement, all variables and given/known data Prove that any group with three elements is isomorphic to $$Z_{3}$$ 2. Relevant equations 3. The attempt at a solution Let G be the group of three elements We have an isomorphism if given c:G--->$$Z_{3}$$, if c is one-to -one and onto and c(ab)=c(a)c(b) First, we check one-to-one We want c(a)=c(b) to imply a=b My problem here is how to define c(a), c(b). Onto: We want c(a)=x and want to solve for a? c(ab): Same problem with not knowing what c(ab) is
 Mentor P: 18,036 You assumed that c(a)=c(b), and from that assimption followed that 1=2. So your assumption is wrong, and thus $$c(a)\neq c(b)$$