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global $k$-group; $\Sigma$-isotype subgroup; $\ast$-isotype subgroup; knice subgroup; primitive element; $\ast$-valuated coproduct
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.
Hill P., Megibben C.: Torsion free groups. Trans. Amer. Math. Soc. 295 (1986), 735-751. MR 0833706 | Zbl 0597.20047
Hill P., Megibben C.: Knice subgroups of mixed groups. Abelian Group Theory Gordon-Breach New York (1987), 89-109. MR 1011306 | Zbl 0653.20057
Hill P., Megibben C.: Pure subgroups of torsion-free groups. Trans. Amer. Math. Soc. 303 (1987), 765-778. MR 0902797 | Zbl 0627.20028
Hill P., Megibben C.: Mixed groups. Trans. Amer. Math. Soc. 334 (1992), 121-142. MR 1116315 | Zbl 0798.20050
Hill P., Megibben C., Ullery W.: $\Sigma$-isotype subgroups of local $k$-groups. Contemp. Math. 273 (2001), 159-176. MR 1817160 | Zbl 0982.20038
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