THE SYLOW STRUCTURE OF SCALAR AUTOMORPHISM GROUPS

We shall review basically known facts about periodic locally compact abelian groups. For any periodic locally compact abelian group $A$, its automorphism group contains (as a subgroup) those automorphisms that leave invariant every closed subgroup of $A$; to be denoted by $\mathrm{SAut}(A)$ . This subgroup is profinite in the $g$-Arens topology and hence allows a decomposition into its $p$-primary subgroups for the primes $p$ for which topological $p$-elements in this automorphism subgroup exist. The interplay between the $p$-primary decomposition of $\mathrm{SAut}(A)$ and $A$ can be encoded in a bipartite graph, the mastergraph of $A$. Properties and applications of this concept are discussed.