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.
@article{119257,
author = {Charles K. Megibben and William Ullery},
title = {Isotype subgroups of mixed groups},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {42},
year = {2001},
pages = {421-442},
zbl = {1102.20037},
mrnumber = {1859590},
language = {en},
url = {http://dml.mathdoc.fr/item/119257}
}
Megibben, Charles K.; Ullery, William. Isotype subgroups of mixed groups. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) pp. 421-442. http://gdmltest.u-ga.fr/item/119257/
Torsion free groups, Trans. Amer. Math. Soc. 295 (1986), 735-751. (1986) | MR 0833706 | Zbl 0597.20047
Knice subgroups of mixed groups, Abelian Group Theory Gordon-Breach New York (1987), 89-109. (1987) | MR 1011306 | Zbl 0653.20057
Pure subgroups of torsion-free groups, Trans. Amer. Math. Soc. 303 (1987), 765-778. (1987) | MR 0902797 | Zbl 0627.20028
Mixed groups, Trans. Amer. Math. Soc. 334 (1992), 121-142. (1992) | MR 1116315 | Zbl 0798.20050
$\Sigma$-isotype subgroups of local $k$-groups, Contemp. Math. 273 (2001), 159-176. (2001) | MR 1817160 | Zbl 0982.20038