Clarify: "$S^0$, $S^1$ and $S^3$ are the only spheres which are also groups" The zero, one, and three dimensional spheres $S^0$, $S^1$ and $S^3$ are in bijection with the sets $\{a\in \mathbb{K}:|a|=1\}$ for $\mathbb{K} = \mathbb{R}, \mathbb{C}, \mathbb{H}$ respectively.
The real, complex and quaternionic multiplication therefore provide a group operation on these spheres.
This is mentioned in the book: Kristopher Tapp (2011), Matrix Groups for Undergraduates, Indian Edition, pp. 40.
Following this there is a statement:

It turns out that $S^0$, $S^1$ and $S^3$ are the only spheres which are also groups.

Can someone please clarify this statement?
How are these three the only spheres which are also groups?
For example, I could take any sphere $S^k$ ($k\ge 1$) and get a bijection $f:S^k \to S^1$ and define a binary operation on $S^k$ by $$a*b = f^{-1}(f(a)\cdot f(b))$$
and $S^k$ would be a group under this operation.
So what exactly is meant by the above statement? In what sense are these the only three spheres which are also groups?

The spheres $S^0,S^1$ and $S^3$ are the only spheres that are lie groups (a group that is a differentiable manifold as well). The proof uses the group cohomology (ie, studying groups using cohomology theory) of spheres. Check out the De Rham cohomology of the $n-$dimensional sphere, which states that $H^1(S^n\times I)$ and $H^3(S^n\times I)$ both equal $0$ as long as $n$ does not equal $1$ or $3$ (in this case). Here, $I$ is a real and open interval (eg, the real numbers). Check out the following links:
http://en.wikipedia.org/wiki/De_Rham_cohomology#The_n-sphere
http://planetmath.org/spheresthatareliegroups
