Commutative modular group algebras of p -mixed and p -splitting abelian Σ -groups

Peter Vassilev Danchev

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 3, page 419-428
  • ISSN: 0010-2628

Abstract

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Let G be a p -mixed abelian group and R is a commutative perfect integral domain of char R = p > 0 . Then, the first main result is that the group of all normalized invertible elements V ( R G ) is a Σ -group if and only if G is a Σ -group. In particular, the second central result is that if G is a Σ -group, the R -algebras isomorphism R A ≅ R G between the group algebras R A and R G for an arbitrary but fixed group A implies A is a p -mixed abelian Σ -group and even more that the high subgroups of A and G are isomorphic, namely, ℋ A ≅ ℋ G . Besides, when G is p -splitting and R is an algebraically closed field of char R = p ≠ 0 , V ( R G ) is a Σ -group if and only if G p and G / G t are both Σ -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.

How to cite

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Danchev, Peter Vassilev. "Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma $-groups." Commentationes Mathematicae Universitatis Carolinae 43.3 (2002): 419-428. <http://eudml.org/doc/248994>.

@article{Danchev2002,
abstract = {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, $\{\mathcal \{H\}\}_A \cong \{\mathcal \{H\}\}_G$. Besides, when $G$ is $p$-splitting and $R$ is an algebraically closed field of $\operatorname\{char\} R = p \ne 0$, $V(RG)$ 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.},
author = {Danchev, Peter Vassilev},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups; group algebras; high subgroups; -mixed Abelian groups; -splitting Abelian groups; Abelian -groups; groups of units; isomorphism problem},
language = {eng},
number = {3},
pages = {419-428},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma $-groups},
url = {http://eudml.org/doc/248994},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Danchev, Peter Vassilev
TI - Commutative modular group algebras of $p$-mixed and $p$-splitting abelian $\Sigma $-groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 3
SP - 419
EP - 428
AB - 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, ${\mathcal {H}}_A \cong {\mathcal {H}}_G$. Besides, when $G$ is $p$-splitting and $R$ is an algebraically closed field of $\operatorname{char} R = p \ne 0$, $V(RG)$ 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.
LA - eng
KW - group algebras; high subgroups; $p$-mixed and $p$-splitting groups; $\Sigma $-groups; group algebras; high subgroups; -mixed Abelian groups; -splitting Abelian groups; Abelian -groups; groups of units; isomorphism problem
UR - http://eudml.org/doc/248994
ER -

References

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  1. Danchev P.V., Units in abelian group rings of prime characteristic, C.R. Acad. Bulgare Sci. 48 (8) (1995), 5-8. (1995) Zbl0852.16024MR1406422
  2. Danchev P.V., Commutative group algebras of σ -summable abelian groups, Proc. Amer. Math. Soc. 125 (9) (1997), 2559-2564. (1997) Zbl0886.16024MR1415581
  3. Danchev P.V., Commutative group algebras of abelian Σ -groups, Math. J. Okayama Univ. 40 (1998), 77-90. (1998) MR1755921
  4. Danchev P.V., Isomorphism of modular group algebras of totally projective abelian groups, Comm. Algebra 28 (5) (2000), 2521-2531. (2000) Zbl0958.20003MR1757478
  5. Danchev P.V., Modular group algebras of coproducts of countable abelian groups, Hokkaido Math. J. 29 (2) (2000), 255-262. (2000) Zbl0967.20003MR1776708
  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. (2001) MR1826044
  7. Danchev P.V., Sylow p -subgroups of modular abelian group rings, C.R. Acad. Bulgare Sci. 54 (2) (2001), 5-8. (2001) Zbl0972.16018MR1826186
  8. Danchev P.V., Sylow p -subgroups of commutative modular and semisimple group rings, C.R. Acad. Bulgare Sci. 54 (6) (2001), 5-6. (2001) Zbl0987.16023MR1845379
  9. Danchev P.V., Normed units in abelian group rings, Glasgow Math. J. 43 (3) (2001), 365-373. (2001) Zbl0997.16019MR1878581
  10. Danchev P.V., Criteria for unit groups in commutative group rings, submitted. Zbl1120.16302
  11. Danchev P.V., Maximal divisible subgroups in modular group algebras of p -mixed and p -splitting abelian groups, submitted. Zbl1086.16017
  12. Danchev P.V., Basic subgroups in abelian group rings, Czechoslovak Math. J. 52 1 (2002), 129-140. (2002) Zbl1003.16026MR1885462
  13. Danchev P.V., Basic subgroups in commutative modular group rings, submitted. Zbl1057.16028
  14. Danchev P.V., Basic subgroups in group rings of abelian groups, J. Group Theory, to appear. MR1826044
  15. Danchev P.V., Commutative modular group algebras of Warfield Abelian groups, Trans. Amer. Math. Soc., to appear. MR2038834
  16. Fuchs L., Infinite Abelian Groups, vol. I and II, Mir, Moscow, 1974 and 1977 (in Russian). Zbl0338.20063MR0457533
  17. Higman G., The units of group rings, Proc. London Math. Soc. 46 (1938-39), 231-248. (1938-39) Zbl0025.24302MR0002137
  18. Irwin J.M., High subgroups of abelian torsion groups, Pacific J. Math. 11 (1961), 1375-1384. (1961) Zbl0106.02303MR0136654
  19. Irwin J.M., Walker E.A., On N -high subgroups of abelian groups, Pacific J. Math. 11 (1961), 1363-1374. (1961) Zbl0106.02304MR0136653
  20. Irwin J.M., Walker E.A., On isotype subgroups of abelian groups, Bull. Soc. Math. France 89 (1961), 451-460. (1961) Zbl0102.26701MR0147539
  21. Irwin J.M., Peercy C., Walker E.A., Splitting properties of high subgroups, Bull. Soc. Math. France 90 (1962), 185-192. (1962) Zbl0106.02401MR0144958

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