On iterated torsion products of Abelian \(p\)-groups
The author considers the group \(\text{Tor}(A,B)\), where \(A\) and \(B\) are \(p\)-groups. The main results show when the group \(\text{Tor}(A,B)\) is a direct sum of countable groups (dsc). Some cases were considered by Nunke and Hill. Let \(\lambda\) be a limit ordinal (\(\lambda\leq\Omega\)), Megibben calls \(G\) a \(C_ \lambda\) group if for every \(\alpha<\lambda\) the group \(G/G(\alpha)\) is a dsc. Theorem 1. Let \(A\) and \(B\) be reduced groups with \(\lambda=\ell(A)<\ell(B)\) (\(\ell(A)\), \(\ell(B)\) are lengths of \(A\) and \(B\)). Then \(\text{Tor}(A,B)\) is a dsc if and only if \(A\) is a dsc and \(B\) is a \(C_ \lambda\) group. Theorem 2. Let \(G\) be a group and \(H\) a reduced dsc of length \(\lambda\). Then \(G\) is \(C_ \lambda\) if and only if \(\text{Tor}(G,H)\) is a dsc. The author introduces a definition which is fundamental for the results of the paper. Let \(G\) be an infinite \(C_ \lambda\) group. By a \(\lambda\)-development of \(G\) we mean an ascending chain of subgroups \(\{G_ i\}_{i\in I}\) of \(G\), where \(I=[0,\alpha]\) for some limit ordinal \(\alpha\) such that we have (a) for all \(i<\alpha\), \(G_ i\) is a \(\lambda\)-pure subgroup of \(G\); (b) \(G_ 0=0\), \(G_ \alpha=G\); (c) if \(i\leq\alpha\) is a limit ordinal, \(\bigcup_{j<i} G_ j=G_ i\). If, in addition to the above, we have (d) \(| G_ i|<| G|\) for all \(i<\alpha\), we will call the \(\lambda\)-development proper. Finally, if in addition to (a)-(d) we have (e) \((G_{i+1}/G_ i)(\lambda)=0\) if and only if \((G/G_ i)(\lambda)=0\), for all \(i<\alpha\) we will call the \(\lambda\)-development normal. Theorem 3. Suppose \(A\) and \(B\) are normal \(C_ \lambda\) groups with \(|\lambda| \leq | A| < | B|\). Then \(\text{Tor}(A,B)\) is a dsc if and only if \(B\) has a \(\lambda\)-development \(\{B_ i\}\) such that, for each \(i\), \(\text{Tor}(A,B_{i+1}/B_ i)\) is a dsc. Furthermore, if \(\text{Tor}(A,B)\) is a dsc, we may choose the \(\lambda\)-development to be normal. In the paper the iterated torsion product is defined inductively by \(\text{Tor}(A_ 1,A_ 2,\dots, A_ n)=\text{Tor}(\text{Tor}(A_ 1,\dots,A_{n-1}),A_ n)\). The main results for the iterated torsion product are as follows. Theorem 5. Suppose \(A_ 1,A_ 2,\dots,A_ n\) are normal \(C_ \lambda\) groups of cardinality at most \(\aleph_{n-1}\). Then \(\text{Tor}(A_ 1,A_ 2,\dots,A_ n)\) is a dsc group. Theorem 8. Suppose \(\lambda\) is countable, \(A_ 1,A_ 2,\dots,A_ n\) are normal \(C_ \lambda\) groups of equal uncountable cardinality. If \(\text{Tor}(A_ 1,A_ 2,\dots,A_ n)\) is a dsc, then there are normal \(\lambda\)-developments \(\{B_{1,i}\},\dots,\{B_{n,i}\}\) such that for all \(j\) and \(i\) \[ Y_{i,j}=\text{Tor}(B_{1,i},\ldots,B_{j- 1,i},B_{j,i+1}/B_{j,i},B_{j+1,i},\ldots,B_{n,i}) \] is a dsc.
scientific article; zbMATH DE number 150393
scientific article