Sylow P-Subgroups of Abelian Group Rings

Abstract

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).