In this lecture we consider the effects of symmetry on bifurcation problems. Even a simple reflectional symmetry can have important consequences. For example, recall the two problems with the elastic from Lectures 1 and 2 which exhibited pitchfork bifurcations. Why should the pitchfork ever be seen? After all, we saw in the last lecture that a bifurcation diagram which contains either a bifurcation point (i.e., a solution of (2.12)) or and the pitchfork has both. How does it happen that a natural mathematical description of these problems predicts this apparently very unstable phenomenon, a pitchfork? The answer lies in the fact that both these problems possessed a nontrivial reflectional symmetry: u → -u for the compression problem of Lecture 1, u(s) → u (π - s) for the shallow arch of Lecture 2. It turns out that the pitchfork is persistent within the class of functions admitting a nontrivial action of the group Z2. (This fact indicates some of the problems of generalizing Theorem 2.2 to cases where a symmetry group acts.) See Golubitsky and Schaeffer [1979b] for a proof of persistence; here we do not pursue it further.