Article
Full entry |
PDF
(0.3 MB)
Feedback
Keywords:
$p$-basic subgroups; normalized units; group algebras; starred groups
$p$-basic subgroups; normalized units; group algebras; starred groups
Summary:
Suppose ${F}$ is a perfect field of ${\mathop {\mathrm char}F=p\ne 0}$ 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 (1996) in Houston J. Math. 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. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.
References:
[1] D. O. Cutler: Another summable $C_\Omega $-group. Proc. Amer. Math. Soc. 26 (1970), 43–44. MR 0262355
[2] P. V. Danchev: Topologically pure and basis subgroups in commutative group rings. Compt. Rend. Acad. Bulg. Sci. 48 (1995), 7–10. MR 1405499 | Zbl 0853.16040
[3] P. V. Danchev: Commutative group algebras of $\sigma $-summable abelian groups. Proc. Amer. Math. Soc. 125 (1997), 2559–2564. DOI 10.1090/S0002-9939-97-04052-5 | MR 1415581 | 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. DOI 10.14492/hokmj/1350911954 | MR 1844820 | Zbl 0989.16019
[5] P. V. Danchev: Basic subgroups in abelian group rings. Czechoslovak Math. J. 52 (2002), 129–140. DOI 10.1023/A:1021779506416 | MR 1885462 | Zbl 1003.16026
[6] P. V. Danchev: Basic subgroups in commutative modular group rings. Math. Bohem. 129 (2004), 79–90. MR 2048788 | Zbl 1057.16028
[7] P. V. Danchev: Subgroups of the basic subgroup in a modular group ring. Math. Slovaca 55 (2005), 431–441. MR 2181782 | 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. MR 1845379 | Zbl 0987.16023
[9] L. Fuchs: Infinite abelian groups, I. Mir, Moscow, 1974. (Russian) MR 0346073
[10] P. D. Hill: A summable $C_{\Omega }$-group. Proc. Amer. Math. Soc. 23 (1969), 428–430. MR 0245674
[11] G. Karpilovsky: Unit groups of group rings. North-Holland, Amsterdam, 1989. MR 1042757 | Zbl 0687.16010
[12] L. Kovács: On subgroups of the basic subgroup. Publ. Math. Debrecen 5 (1958), 261–264. MR 0100628
[13] W. May: The direct factor problem for modular abelian group algebras. Contemp. Math. 93 (1989), 303–308. DOI 10.1090/conm/093/1003359 | MR 1003359 | Zbl 0676.16010
[14] W. May: Modular group algebras of simply presented abelian groups. Proc. Amer. Math. Soc. 104 (1988), 403–409. DOI 10.1090/S0002-9939-1988-0962805-2 | MR 0962805 | 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. MR 1402745
Search
Partner of