Homogeneous linearly ordered spaces

Every compact subset of a homogeneous generalized ordered (GO) space has character at most ${\omega }_1$ and cardinality at most $2^{{\omega }_1}$; if such a subset has uncountable character, then the character of the whole space equals ${\omega }_1$ and its $\pi$-character is countable. We construct a homogeneous $\sigma$-compact linearly ordered space (LOTS) $\mathbf{H}$ containing a compact subset $\mathbf{S}$ of cardinality $2^{{\omega }_1}$ whose character is ${\omega }_1$ at every point and whose weight and Souslin number are both $2^{{\omega }_1}$; thus both bounds obtained are sharp. We prove that a semitopological group that is a GO space is hereditarily paracompact; if, in addition, it is not a $P$-space, then it is submetrizable, has countable character, and its compact subsets are metrizable. Every linearly ordered semitopological group (and, more generally, every GO semitopological group) is either metrizable or is a $P$-space; the same holds for topological groups. We also show that in an order-homogeneous LOTS every compact subset is first countable.