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Keywords:
group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups
group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups
Summary:
Let $G$ be a $p$-mixed abelian group and $R$ is a commutative perfect integral domain of $\operatorname{char} R = p > 0$. Then, the first main result is that the group of all normalized invertible elements $V(RG)$ is a $\Sigma $-group if and only if $G$ is a $\Sigma $-group. In particular, the second central result is that if $G$ is a $\Sigma $-group, the $R$-algebras isomorphism $RA\cong RG$ between the group algebras $RA$ and $RG$ for an arbitrary but fixed group $A$ implies $A$ is a $p$-mixed abelian $\Sigma $-group and even more that the high subgroups of $A$ and $G$ are isomorphic, namely, ${\Cal H}_A \cong {\Cal H}_G$. Besides, when $G$ is $p$-splitting and $R$ is an algebraically closed field of $\operatorname{char} R = p \not= 0$, $V(RG)$ is a $\Sigma $-group if and only if $G_p$ and $G/G_t$ are both $\Sigma $-groups. These statements combined with our recent results published in Math. J. Okayama Univ. (1998) almost exhausted the investigations on this theme concerning the description of the group structure.
References:
[1] Danchev P.V.: Units in abelian group rings of prime characteristic. C.R. Acad. Bulgare Sci. 48 (8) (1995), 5-8. MR 1406422 | Zbl 0852.16024
[2] Danchev P.V.: Commutative group algebras of $\sigma$-summable abelian groups. Proc. Amer. Math. Soc. 125 (9) (1997), 2559-2564. MR 1415581 | Zbl 0886.16024
[3] Danchev P.V.: Commutative group algebras of abelian $\Sigma$-groups. Math. J. Okayama Univ. 40 (1998), 77-90. MR 1755921
[4] Danchev P.V.: Isomorphism of modular group algebras of totally projective abelian groups. Comm. Algebra 28 (5) (2000), 2521-2531. MR 1757478 | Zbl 0958.20003
[5] Danchev P.V.: Modular group algebras of coproducts of countable abelian groups. Hokkaido Math. J. 29 (2) (2000), 255-262. MR 1776708 | Zbl 0967.20003
[6] Danchev P.V.: Unit groups and Sylow $p$-subgroups in commutative group rings of prime characteristic $p$. C.R. Acad. Bulgare Sci. 54 (1) (2001), 7-10. MR 1826044
[7] Danchev P.V.: Sylow $p$-subgroups of modular abelian group rings. C.R. Acad. Bulgare Sci. 54 (2) (2001), 5-8. MR 1826186 | Zbl 0972.16018
[8] Danchev P.V.: Sylow $p$-subgroups of commutative modular and semisimple group rings. C.R. Acad. Bulgare Sci. 54 (6) (2001), 5-6. MR 1845379 | Zbl 0987.16023
[9] Danchev P.V.: Normed units in abelian group rings. Glasgow Math. J. 43 (3) (2001), 365-373. MR 1878581 | Zbl 0997.16019
[10] Danchev P.V.: Criteria for unit groups in commutative group rings. submitted. Zbl 1120.16302
[11] Danchev P.V.: Maximal divisible subgroups in modular group algebras of $p$-mixed and $p$-splitting abelian groups. submitted. Zbl 1086.16017
[12] Danchev P.V.: Basic subgroups in abelian group rings. Czechoslovak Math. J. 52 1 (2002), 129-140. MR 1885462 | Zbl 1003.16026
[13] Danchev P.V.: Basic subgroups in commutative modular group rings. submitted. Zbl 1057.16028
[14] Danchev P.V.: Basic subgroups in group rings of abelian groups. J. Group Theory, to appear. MR 1826044
[15] Danchev P.V.: Commutative modular group algebras of Warfield Abelian groups. Trans. Amer. Math. Soc., to appear. MR 2038834
[16] Fuchs L.: Infinite Abelian Groups. vol. I and II, Mir, Moscow, 1974 and 1977 (in Russian). MR 0457533 | Zbl 0338.20063
[17] Higman G.: The units of group rings. Proc. London Math. Soc. 46 (1938-39), 231-248. MR 0002137 | Zbl 0025.24302
[18] Irwin J.M.: High subgroups of abelian torsion groups. Pacific J. Math. 11 (1961), 1375-1384. MR 0136654 | Zbl 0106.02303
[19] Irwin J.M., Walker E.A.: On $N$-high subgroups of abelian groups. Pacific J. Math. 11 (1961), 1363-1374. MR 0136653 | Zbl 0106.02304
[20] Irwin J.M., Walker E.A.: On isotype subgroups of abelian groups. Bull. Soc. Math. France 89 (1961), 451-460. MR 0147539 | Zbl 0102.26701
[21] Irwin J.M., Peercy C., Walker E.A.: Splitting properties of high subgroups. Bull. Soc. Math. France 90 (1962), 185-192. MR 0144958 | Zbl 0106.02401
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