Homogeneous Locally Compact Groups Original Research Article

We determine all locally compact abelian groups with the property that the group of all topological automorphisms acts transitively on the set of nontrivial elements. Such groups are called homogeneous. The connected ones are the additive groups of finite-dimensional vector spaces over the real numbers. The compact ones are the (not necessarily finite) powers of cyclic groups of prime order. Actually, the commutativity hypothesis is needed only in the remaining cases: the disconnected torsion-free homogeneous abelian locally compact groups are the divisible hulls of powers of the group ofp-adic integers; and the homogeneous abelian locally compact torsion groups are the products of (compact) powers of cyclic groups and discrete elementary abelian groups. A characterization of additive groups of vector spaces of finite dimension over locally compact fields is obtained.