A Lie group G is a manifold that is also a group *such that* the group operations of multiplication : G x G —> G and inversion G —> G are each continuous. The solution of Hilbert's Fifth Problem showed that any such Lie group is in fact a *real-analytic* manifold on which those group operations are in fact real analytic. (Meaning: expressible locally as a tuple of power series.) A few sweeping things are known about Lie groups. For every dimension of Euclidean space R^n there is the Lie group SO(n) of all its isometries. (The rotation group SO(n) is the connected component of the identity element in O(n).) Every compact Lie group is a Lie subgroup of an O(n) for some n, so in that sense the O(n)'s are "universal". The rotation group SO(n) is abelian only for n = 1 or n = 2. The Lie groups that are abelian are precisely the cartesian products of Euclidean spaces and tori: R^n x T^p (where the p-torus T^p is the cartesian product of p circles S^1). The homotopy groups π_n(X) of a path-connected space X are the continuous maps of the n-sphere S^n into X, factored out by the equivalence relation of being continuously deformable into each other ("homotopy"). If we let each R^n sit in the next one as R^n x {0} in R^(n+1), then the orthogonal groups O(n) form an increasing union. We can then take the *union* of all the O(n)'s — to get a group (that isn't Lie) called O(oo). Here are the homotopy groups of O(oo), which striking depend only on n mod 8: n mod 8 0 1 2 3 4 5 6 7 π_n(O(oo)) Z/2 Z/2 0 Z 0 0 0 Z This was proved by Bott in 1957, and is called Bott periodicity. —Dan