This paper is centred around a single question: can a minimal left ideal L in G[Lscr ][Uscr ][Cscr ], the largest semi-group compactification of a locally compact group G, be itself algebraically a group? Our answer is no (unless G is compact). In deriving this conclusion, we obtain for nearly all groups the stronger result that no maximal subgroup in L can be closed. A feature of our work is that completely different techniques are required for the connected and totally disconnected cases. For the former, we can rely on the extensive structure theory of connected, non-compact, locally compact groups to derive 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 other 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 minimal left ideals in G[Lscr ][Uscr ][Cscr ] and H[Lscr ][Uscr ][Cscr ] 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 use our methods to show that there is no finite subset of G[Lscr ][Uscr ][Cscr ] invariant under the right action of G when G is an almost connected group or an IN-group.