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# Article

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Keywords:
global \$k\$-group; \$\Sigma\$-isotype subgroup; \$\ast\$-isotype subgroup; knice subgroup; primitive element; \$\ast\$-valuated coproduct
Summary:
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.
References:
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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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