I know that some $ S^n $ s have group structures, whereas others do not.
For example,
$ S^0 = \{\pm 1\} \cong \mathbb{Z}/2\mathbb{Z}$,
$ S^1 = \mathbb{R}/\mathbb{Z} $,
$ S^3 = \{a+bi+cj+dk:a^2+b^2+c^2+d^2=1\} $, a multiplicative subgroup of the real quaternion algebra.
On the other hand, $ S^{2n} $ with $ n \geq 1 $ cannot have a group structure. Proposition 2.29 of A. Hatcher's Algebraic Topology (on page 135) states that $ \mathbb{Z}/2\mathbb{Z} $ is the only nontrivial group that can act freely on $ S^{2n} $. But if $ S^{2n} $ itself was a group, then it acts freely on itself.
I want to know whether there is a complete classification theorem about for which $ n $ does $ S^n $ admit a group structure (not necessarily a Lie group structure). Thanks in advance!