Sylow P-Subgroups of Abelian Group Rings

Danchev, P.. "Sylow P-Subgroups of Abelian Group Rings." Serdica Mathematical Journal 29.1 (2003): 33-44. <http://eudml.org/doc/219614>.

@article{Danchev2003,
abstract = {2000 Mathematics Subject Classification: Primary 20C07, 20K10, 20K20, 20K21; Secondary 16U60, 16S34.Let PG be the abelian modular group ring of the abelian group G over the abelian ring P with 1 and prime char P = p. In the present article,the p-primary components Up(PG) and S(PG) of the groups of units U(PG) and V(PG) are classified for some major classes of abelian groups. Suppose K is a first kind field with respect to p in char K ≠ p and A is an abelian p-group. In the present work, the p-primary components Up(KA) and S(KA) of the group of units U(KA) and V(KA) in the semisimple abelian group ring KA are studied when they belong to some central classes of abelian groups. The established criteria extend results obtained by us in Compt. rend. Acad. bulg. Sci. (1993). Moreover, the question for the isomorphic type of the basic subgroup of S(KA) is also settled. As a final result, it is proved that if A is a direct sum of cyclics, the group of all normed p-units S(KA) modulo A, that is, S(KA)/A, is a direct sum of cyclics too. Thus A is a direct factor of S(KA) with a direct sum of cyclics complementary factor provided sp(K) ⊇ N. This generalizes a result due to T. Mollov in Pliska Stud. Math. Bulgar. (1986).},
author = {Danchev, P.},
journal = {Serdica Mathematical Journal},
keywords = {Unit Groups; Direct Factors; Basic Subgroups; Direct Sums of Cyclics; unit groups; direct factors; basic subgroups; direct sums of cyclic groups},
language = {eng},
number = {1},
pages = {33-44},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {Sylow P-Subgroups of Abelian Group Rings},
url = {http://eudml.org/doc/219614},
volume = {29},
year = {2003},
}

TY - JOUR
AU - Danchev, P.
TI - Sylow P-Subgroups of Abelian Group Rings
JO - Serdica Mathematical Journal
PY - 2003
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 29
IS - 1
SP - 33
EP - 44
AB - 2000 Mathematics Subject Classification: Primary 20C07, 20K10, 20K20, 20K21; Secondary 16U60, 16S34.Let PG be the abelian modular group ring of the abelian group G over the abelian ring P with 1 and prime char P = p. In the present article,the p-primary components Up(PG) and S(PG) of the groups of units U(PG) and V(PG) are classified for some major classes of abelian groups. Suppose K is a first kind field with respect to p in char K ≠ p and A is an abelian p-group. In the present work, the p-primary components Up(KA) and S(KA) of the group of units U(KA) and V(KA) in the semisimple abelian group ring KA are studied when they belong to some central classes of abelian groups. The established criteria extend results obtained by us in Compt. rend. Acad. bulg. Sci. (1993). Moreover, the question for the isomorphic type of the basic subgroup of S(KA) is also settled. As a final result, it is proved that if A is a direct sum of cyclics, the group of all normed p-units S(KA) modulo A, that is, S(KA)/A, is a direct sum of cyclics too. Thus A is a direct factor of S(KA) with a direct sum of cyclics complementary factor provided sp(K) ⊇ N. This generalizes a result due to T. Mollov in Pliska Stud. Math. Bulgar. (1986).
LA - eng
KW - Unit Groups; Direct Factors; Basic Subgroups; Direct Sums of Cyclics; unit groups; direct factors; basic subgroups; direct sums of cyclic groups
UR - http://eudml.org/doc/219614
ER -