why is $S^2$ not a Lie group?

I'm reading John Stillwell's "Naive Lie Theory" and it was mentioned there (without giving a proper definition of what a Lie group is) that the only Lie groups among the unit n-spheres are $S^1$ and $S^3$.
Is there a simple or naive explanation of what makes $S^2$ so different from $S^1$ and $S^3$?

Solution 1:

Every continuos map $S^2\to S^2$ has at least two fixpoints. But for a group element $\ne 1$, left multiplication has no fixpoints.

Solution 2:

As indicated by the comments on Hagen's answer, there are continuous self maps of $S^2$ with no fixed points (e.g., the antipodal map as Chris Z points outs), and orientation preserving homeomorphisms with only one fixed point (e.g., Mobius transformations as Max points out).

As Hagen rightly points out, on a Lie group, there are homeomorphisms which are homotopic to the identity, but which have no fixed points: left multiplication by any non-identity element which lies in the identity component of the Lie group is an example.

So, to show $S^2$ is not a Lie group, it is sufficient to show that every homeomorphism of $S^2$ which is homotopic to the identity has a fixed point. In fact, any continuous map (be it a homeomorphism or not) of $S^2$ has this property. this is proved in this MSE question. In fact, both Bruno Joyal's and Gyu Eun Lee's answers to that question generalize to prove that $S^{2n}$ is not a Lie group for any $n > 0$.