Homogeneous locally compact groups
Motivated by their occurrence in topological geometry, the author studies locally compact groups that are homogeneous in the sense that their automorphism group acts transitively on the non-identity elements. The main results say that compact homogeneous groups are precisely the arbitrary powers of cyclic groups of prime order (with product topology), that connected, locally compact, homogeneous groups are precisely the (additive groups of) vector spaces of finite dimension over the real numbers, and that abelian, locally compact, homogeneous groups are precisely the following: (1) the vector spaces of finite dimension over a nondiscrete locally compact field, (2) the divisible hulls of arbitrary powers of the group of \(p\)-adic integers, (3) all discrete vector spaces over a prime field (finite or the rationals), (4) the compact groups already mentioned, (5) the direct products of a group of type (3) and a group of type (4), based on the same prime number. -- Among the homogeneous, locally compact abelian groups, those of type (1) are characterized by being of countable weight and neither compact nor discrete.
scientific article; zbMATH DE number 1117780