Abstract:In this paper, we initiate the study of various classes of isotype subgroups of global mixed groups. Our goal is to advance the theory of $\Sigma $-isotype subgroups to a level comparable to its status in the simpler contexts of torsion-free and $p$-local mixed groups. Given the history of those theories, one anticipates that definitive results are to be found only when attention is restricted to global $k$-groups, the prototype being global groups with decomposition bases. A large portion of this paper is devoted to showing that primitive elements proliferate in $\Sigma $-isotype subgroups of such groups. This allows us to establish the fundamental fact that finite rank $\Sigma $-isotype subgroups of $k$-groups are themselves $k$-groups.
Keywords: global $k$-group, $\Sigma $-isotype subgroup, $\ast $-isotype subgroup, knice subgroup, primitive element, $\ast $-valuated coproduct
AMS Subject Classification: 20K21, 20K27