Direct Decompositions of LCA Groups
A closed subgroup H of an LCA (locally compact abelian) group G is called topologically pure if nH¯=H∩nG¯" role="presentation" style="box-sizing: border-box; display: inline-table; line-height: normal; letter-spacing: normal; word-spacing: normal; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; position: relative;">nH¯¯¯¯¯¯¯¯=H∩nG¯¯¯¯¯¯¯nH¯=H∩nG¯ for all positive integers n. It is shown that a finite subgroup of an LCA group is a splitting subgroup if and only if it is topologically pure. A characterization of LCA groups with splitting subgroup of all compact elements is established, and the groups splitting in every LCA group in which they are contained as the subgroup of all compact elements are described. The abelian case of Lee’s  Theorem on supplements for the identity component in locally compact groups is proved. An example of a compact abelian group G with nonsplitting identity component G 0 is given, and a supplement for G 0 is determined.
Loth P. (1999) Direct decompositions of LCA groups. In P. C. Eklof & R. Göbel (Eds.). Abelian groups and modules. Trends in Mathematics series. Birkhäuser. Doi:10.1007/978-3-0348-7591-2_24