## Sylow $$p$$-subgroups of Abelian group rings.(English)Zbl 1035.16025

Let $$KG$$ be the group ring of an Abelian group $$G$$ over a commutative ring $$K$$ and let $$V(KG)$$ be the group of normalized units (i.e. the group of augmentation 1) in $$KG$$. In the paper some group-theoretic properties of the $$p$$-component $$S(KG)$$ of $$V(KG)$$ are proved when either (i) $$K$$ is a ring of prime characteristic $$p$$ or (ii) $$K$$ is a field of the first kind with respect to $$p$$ and the characteristic of $$K$$ is different from $$p$$. The author considers case (ii) under the following restriction: the spectrum of $$K$$ with respect to $$p$$ contains all naturals. The considered properties are very general and do not give the structure of $$S(KG)$$.
The most interesting statement is Theorem 7 for the semisimple case, i.e. for case (ii). It states that if $$A$$ is a direct sum of cyclic $$p$$-groups then $$A$$ is a direct factor of $$S(KA)$$, and consequently of $$V(KA)$$. But a full description of $$S(KA)$$, to within isomorphism, is given by the reviewer [PLISKA, Stud. Math. Bulg. 8, 34-46 (1986; Zbl 0662.16008) and ibid. 8, 54-64 (1986; Zbl 0655.16004)]. Therefore Theorem 7 does not have a big value. Besides, the proof is incomplete, since it is based on Lemma 6, in which it is not proved that the second factor in the decomposition of $$S(RG)$$ is a subgroup of $$S(RG)$$. We note that this fact is obvious only in the modular case.
The second proof of Proposition 1 is not true. We can give the following counterexample to this proof. Let $$A$$ be the direct product of the groups $$H$$ and $$R$$, where $$H$$ is an unbounded direct sum of cyclic $$p$$-groups and $$R$$ is a cyclic group of order $$p$$. Then $$A^p=H$$ holds, i.e. $$t=1$$. We choose $$H_k=H$$ for every natural $$k$$. Then the indicated cross-section on p. 35, line 6(–), is not 1, which contradicts the author’s statement.
The proofs of properties (4) and (5) in Theorem 4 are not true, since the decompositions, which are indicated, are valid only when the group $$A$$ is infinite. Theorem 5 is superfluous, since it can be trivially obtained from Theorem 4.
The author discusses on page 45 a result of Nachev’s and at the same time he is citing his own paper. This is incorrect.
The paper is not written clearly. For example, in the cited Theorem [4] at the beginning of section 2, he does not define the group $$V(PG;H)$$, although he preliminarily writes “for the convenience of the reader …”. At the same time it is not understandable how Theorem [4] is used in the proof of Proposition 2. In the proof of Theorem 7 the author uses, without citing, that $$S(KH_n)$$ has finite exponent $$n$$, which is the reviewer’s result (see the above cited papers) and it is not obvious.

### MSC:

 16U60 Units, groups of units (associative rings and algebras) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16S34 Group rings 20E07 Subgroup theorems; subgroup growth 20K21 Mixed groups 20K25 Direct sums, direct products, etc. for abelian groups

### Citations:

Zbl 0662.16008; Zbl 0655.16004
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