## Basic subgroups in modular Abelian group algebras.(English)Zbl 1167.16026

Summary: Suppose $$F$$ is a perfect field of $$\text{char\,}F=p\neq 0$$ and $$G$$ is an arbitrary Abelian multiplicative group with a $$p$$-basic subgroup $$B$$ and $$p$$-component $$G_p$$. Let $$FG$$ be the group algebra with normed group of all units $$V(FG)$$ and its Sylow $$p$$-subgroup $$S(FG)$$, and let $$I_p(FG;B)$$ be the nilradical of the relative augmentation ideal $$I(FG;B)$$ of $$FG$$ with respect to $$B$$.
The main results that motivate this article are that $$1+I_p(FG;B)$$ is basic in $$S(FG)$$, and $$B(1+I_p(FG;B))$$ is $$p$$-basic in $$V(FG)$$ provided $$G$$ is $$p$$-mixed. These achievements extend in some way a result of N. Nachev when $$G$$ is $$p$$-primary. Thus the problem of obtaining a ($$p$$-)basic subgroup in $$FG$$ is completely resolved provided that the field $$F$$ is perfect.
Moreover, it is shown that $$G_p(1+I_p(FG;B))/G_p$$ is basic in $$S(FG)/G_p$$, and $$G(1+I_p(FG;B))/G$$ is basic in $$V(FG)/G$$ provided $$G$$ is $$p$$-mixed. As consequences, $$S(FG)$$ and $$S(FG)/G_p$$ are both starred or divisible groups.

### MSC:

 16U60 Units, groups of units (associative rings and algebras) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 20K27 Subgroups of abelian groups 20K10 Torsion groups, primary groups and generalized primary groups 20K20 Torsion-free groups, infinite rank 20K21 Mixed groups 16S34 Group rings
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### References:

 [1] D. O. Cutler: Another summable C {$$\Omega$$}-group. Proc. Amer. Math. Soc. 26 (1970), 43–44. [2] P. V. Danchev: Topologically pure and basis subgroups in commutative group rings. Compt. Rend. Acad. Bulg. Sci. 48 (1995), 7–10. · Zbl 0853.16040 [3] P. V. Danchev: Commutative group algebras of {$$\sigma$$}-summable abelian groups. Proc. Amer. Math. Soc. 125 (1997), 2559–2564. · Zbl 0886.16024 [4] P. V. Danchev: C {$$\lambda$$}-groups and {$$\lambda$$}-basic subgroups in modular group rings. Hokkaido Math. J. 30 (2001), 283–296. · Zbl 0989.16019 [5] P. V. Danchev: Basic subgroups in abelian group rings. Czechoslovak Math. J. 52 (2002), 129–140. · Zbl 1003.16026 [6] P. V. Danchev: Basic subgroups in commutative modular group rings. Math. Bohem. 129 (2004), 79–90. · Zbl 1057.16028 [7] P. V. Danchev: Subgroups of the basic subgroup in a modular group ring. Math. Slovaca 55 (2005), 431–441. · Zbl 1112.16030 [8] P. V. Danchev: Sylow p-subgroups of commutative modular and semisimple group rings. Compt. Rend. Acad. Bulg. Sci. 54 (2001), 5–6. · Zbl 0987.16023 [9] L. Fuchs: Infinite abelian groups, I. Mir, Moscow, 1974. (In Russian.) · Zbl 0274.20067 [10] P. D. Hill: A summable C {$$\Omega$$}-group. Proc. Amer. Math. Soc. 23 (1969), 428–430. [11] G. Karpilovsky: Unit groups of group rings. North-Holland, Amsterdam, 1989. · Zbl 0687.16010 [12] L. Kovács: On subgroups of the basic subgroup. Publ. Math. Debrecen 5 (1958), 261–264. [13] W. May: The direct factor problem for modular abelian group algebras. Contemp. Math. 93 (1989), 303–308. · Zbl 0676.16010 [14] W. May: Modular group algebras of simply presented abelian groups. Proc. Amer. Math. Soc. 104 (1988), 403–409. · Zbl 0691.20008 [15] N. Nachev: Basic subgroups of the group of normalized units in modular group rings. Houston J. Math. 22 (1996), 225–232. · Zbl 0859.16025
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