On the structure of minimal left ideals in the largest compactification ofa locally compact group

\This paper is centred around a single question: can a minimal left ideal L in G(LUC), the largest semigroup compactification of a locally compact gro up G, be itself algebraically a group? Our answer is no (unless G is compac t). In deriving this conclusion, we obtain for nearly all groups the strong er result that no maximal subgroup in L can be closed. A feature of our wor k is that completely different techniques are required for the connected an d totally disconnected cases. For the former, we can rely on the extensive structure theory of connected, non-compact, locally compact groups to deriv e the solution from the commutative case, using some reduction lemmas. The latter directly involves topological dynamics; we construct a compact space and an action of G on it which has pathological properties. We obtain othe r results as tools towards our main goal or as consequences of our methods. Thus we find an extension to earlier work on the relationship between mini mal left ideals in G(LUC) and H-LUC when H is a closed subgroup of G with G /H compact. We show that the distal compactification of G is finite if and only if the almost periodic compactification of G is finite. Finally, we us e our methods to show that there is no finite subset of G(LUC) invariant un der the right action of G when G is an almost connected group or an IN-grou p.